I 'J 
 
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Cornell University 
 Library 
 
 The original of tliis book is in 
 tine Cornell University Library. 
 
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 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031265295 
 
ComeH Unwerslty Library 
 
AN 
 
 ELEMENTARY TREATISE 
 
 oir 
 
 MECHANICS, 
 
 POK THE USE OP THE JUNIOK CLASSES AT THE UNIVERSITY 
 AND THE HIGHER CLASSES IN SCHOOLS. 
 
 % €alhdxon af ^mm^lm. 
 
 BY 
 
 S. PARKINSON, B.D. 
 
 FELLOW AND PRaLBCTOK OF ST JOHN'S COLLEGE, CAMBBIDQE. 
 
 THIRD EDITION REVISED. 
 
 MACMILLAN AND CO. 
 1863. 
 
UMVERS2TY 
 
 X 
 
 LIBRARY. 
 
 / 
 
 (ffamirfirse: 
 
 PKINTED BY 0. J. CLAY, JI.A. 
 AT THE TJNIVERSITT PKESB. 
 
PEEFACE TO THE THIRD EDITION. 
 
 In preparing a third edition of tliis work I have kept the 
 same object in view as I had in the former editions, — ^namely, 
 to include in it such portions of Theoretical Mechanics as 
 can he conveniently investigated without the use of the 
 Differential Calculus, and so render it suitable as a manual 
 for the Junior Classes in the University and the Higher 
 Classes in Schools. With one or two short exceptions, the 
 Student is not presumed to require a knowledge of any 
 branches of Mathematics beyond the elements of Algebra, 
 Geometry and Trigonometry. 
 
 Motion on a Curve, which is treated of in the last 
 Chapter of the Dynamics, does not seem to admit of any 
 complete discussion without the aid of the. Differential Cal- 
 culus; but in consequence of the present requirements of 
 the Senate-House Examinations, I have put together those 
 theorems respecting cycloidal oscillations and curvilinear 
 motion which admit of a tolerably simple Geometrical ex- 
 position. 
 
 Several additional propositions have been incorporated in 
 the work for the pm-pose of rendering it more complete :— 
 and the collection of Examples and Problems has been 
 largely increased : — to most of them I have annexed results, 
 which I hope will render the collection more useful both to 
 tutor and pupil. 
 
 St John's College, 
 Nov. 1863. 
 
CONTENTS. 
 
 STATICS. 
 
 CHAPTER I. 
 Introduction 
 
 PAGE 
 
 I 
 
 Def. of the terms parUck, rigid, Art. i ; motion, absolute and relative, 2 ; 
 def. of force, line of action, egmlibnum, 3 ; preim/re, tendon, 4 ; mass, 7; 
 mode of comparing forces, standard of weight, 8 — 11; principle of 
 the transmission of force, 11 — 14. 
 
 CHAPTEE II. 
 
 Of forces acting in one plane. . , 12 
 
 Def. oi component, resvltant, 15; pwralMogram of forces, 18 — 2,1; triangle 
 of forces, 21 ; Lami's theorem, 23 ; polygon of forces, 25 ; Leibnitz's 
 theorem, 16; converse of parallelogram offerees, 27; resultant of any 
 system of forces in one plane, 28; of two parallel forces, 29, 30; 
 moment of a force, 31 ; moment of two or more forces equal to that 
 of their resultant, 32 — 34; def. of a fulcrum, a, lever, 35; prindple 
 oftJie lever, $6'; condition of equilibrium of a, system of forces in one 
 plane, 38 — 40; remarks on the mmtiml action of smooth and rough 
 surfaces, 41, 42; tension of strings, 43; conditions of eqmlibrium of 
 three forces, 45; problems, 46 — 51. 
 
 CHAPTEE III. 
 
 Friction 54 
 
 Explanation of the action of friction, 52, £3; laws of friction, 54 j rolling 
 friction, &c. £5; practical method of finding the coefficient of fric- 
 tion, 57; 
 
VI CONTENTS. 
 
 CHAPTER IV. 
 
 PAGB 
 
 Of forces f the directions of which meet in a point ; — tension of 
 
 strings on smooth and rough swrfaces . . 60 
 
 ParaUelopiped of forces, 60; resultant of any system of forces acting at a 
 point, 62; conditions of equUibrium, 63; tension of a string passing 
 over a smooth surface, 65; . . . over a rough surface, 66; ihe funi- 
 cular polygon and catenary 67, 68, 
 
 CHAPTER V. 
 
 Of ihe centre of gramiiy . . .73 
 
 Def. of vertical, horizontal, cerUre of gravity, 6g ; there is one and only 
 one centre of gravity for any system, 70 ; centre of parallel forces, 70, 
 Cor. 3 ; centre of gravity of a right line, a paraUelogram, a triangle, 
 perimeter of a triangle, 72, 73 ; centre of gravity of a system of 
 particles in a line, in a plane, arranged in any manner, 74; def. of 
 moment of a force with respect to a plane, centre of mean position, 
 centre of figure, 75; general remarks, 77, 7^; centre of gravity of a 
 pyramid^, 79; stable and unstable equilibrium, 80 — 84; centre of 
 gravity of a circula/r are, sector and segment, 85, 86 ; Leibnitz's 
 Theorem, 87. 
 
 CHAPTER VL 
 
 Meehamical powers . . .106 
 
 The lever discussed, 88 — 93; the commmi steelyard, 94; Danish sleeh/ard, 
 95; comrrum balance, 96 — 98; Balance of Quintenz, 99; wheel and 
 asde, 100, loi; single pvily, 102, .103; systems oi puUies, 104—106; 
 Spanish Barton, 107; inclined plane, smooth and rough, 108 — no; 
 screw. III — 114; wedge, 115; explanation of the principle of virtual 
 velocities, 116 — 118; the principle applied to the several mechanical 
 powers, 119 — 17,8; mecJumieal advantage and effidency, 129; labowr- 
 ing force, what is gained in power is lost in vdodly, horse-power, 
 I3i> '3*5 differential axle, 133; Hunter's Screw, 134. 
 
CONTENTS. vil 
 
 DYNAMICS. 
 
 CHAPTEB I. 
 
 PAGE 
 
 Introduction . . , . 152 
 
 Def. and measure of velocity, Art. 2 ; formula for uniform motion, 4 ; 
 measure of acceleration, J, 6; mass, 8 — 11; momentum, 12; mov- 
 ing fm-ce, 15; impidsive force, 17, 18; geometrical resolution and 
 composition 'of velocities, and accelerations, 19, 20 ; pa/rallelogram 
 of velocities and accelerations, 11 — 25; f/rst law of mMion, 27, ■28; 
 second law of mAjtion, 29 — 33 ; dynamical parallelogram of velocities, 
 35 ; relative motion, 40 ; third law of motion, 42 — 45 ; aclirni and 
 reaction, 46, 47. 
 
 CHAPTER II. 
 
 Of ti/niform metion and collision . . 18 r 
 
 Relative motion of two points or balls, £0, 51; def. of elasticity, force of 
 restitution, modulus of elasticity, 52 ; line of impact, 53 ; impact is a 
 pressure of short duration, 54 ; collision of two balls direct and oblique, 
 55 — 59; impact on a plane, 60; motion of the centre of gravity of 
 two balls, 61, 62 ; problems, 63 — 66, 
 
 CHAPTER III, 
 Of uni/ormlr/ accelerated motion . , 202 
 
 Velocity acquired and space described under a uniform force, 67, 68; 
 geometrical illustration, 69, 71; formmlcB, 73; motion of two bodies 
 connected by a string over a puUy or on an inclined plane, 75 — 77; 
 motion up and down an inclined plane, 78, 79; lines of quickest 
 descent, 80; Atwood's machine, 82. 
 
VlU CONTENTS. 
 
 CHAPTER IV. 
 
 PAGE 
 
 Of the motion of projectiles . . , 219 
 
 Path of a projectile, 85 ; range, time of flight, elevation, 85, 86; on an 
 inclined plane, 87 ; path referred to rectangular co-ordinates, 88 ; 
 motion on a smooth inclined plane, 89; problems, and remarks on 
 the imperfection of the theory, 90 — 94. 
 
 CHAPTER V. 
 
 Motion on a curve . . . 235 
 
 Velocity acquired down a curve, 96, 97; on a circle, 97; properties of the 
 cycloid, 99 — loi ; time of falling down an arc of a cycloid, 102 ; 
 oscillation in a cycloid, 102, Cor. ; length of seconds pendulum, 
 103 ; problem, 104 ; normal acceleration of a particle moving in a 
 curve, 105; centrifugal force, 106; pressure on a curve, 107; pro- 
 blems, 108, 109; Newton's method of determining the elasticity of 
 balls, no. 
 
 Problems and Examples ........ 256 — 374 
 
STATICS. 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 1. Mechanics is the science which treats of the laws of 
 rest and motion of matter. 
 
 A general notion, of the meaning of the term matter is 
 acquired in the daily experience of life, since matter in 
 various forms and under various circumstances is perpetually 
 affecting our senses: we shall therefore assume that the 
 notion of it is familiar to the student. 
 
 A particle or material point is a portion of matter inde- 
 finitely small in all its dimensions; so that its length, breadth, 
 and thickness are less than any assignable linear magnitude. 
 A hody of finite size may be regarded as an aggregation of 
 an indefinitely great number of particles ; and the dimensions 
 of any given hody being limited in every direction, it will 
 consequently have a determinate _^r?re and volume. 
 
 A body or system of bodies all the points of which are 
 held together in an invariable position with respect to one 
 another, is said to be rigid. 
 
 2. When a body or particle constantly occupies the same 
 position in space, it is said to be at rest; and when its position 
 in space changes continuously in any manner whatever, it is 
 said to be in motion. All matter is capable of motion, but 
 we can only judge of the state of rest or motion of a particle 
 
 S^fv. M. 1 
 
2 INTEODUCTION. 
 
 hj comparing it with other particles ; for this reason all the 
 motions which we can observe are necessarily relative mo- 
 tions. 
 
 When a great numher of objects maintain the same rela- 
 tive position, our first impression is to consider them as at 
 rest ; and if one of them changes its position relatively to the 
 others, it is to it that we ascribe the motion. Thus for in- 
 stance, the earth was for a long time considered to be fixed in 
 space, notwithstanding the motions of the sun, moon and stars 
 relatively to objects on the earth's surface with which the 
 observer compared them. The motion was ascribed to them 
 whilst the earth was assumed to be fixed. A careful study of 
 natural phenomena may modify this first impression, but we 
 can never arrive at absolute certainty in this respect ; and the 
 conclusions respecting absolute motions, to which we are led 
 by the observation of relative motions, can only be regarded 
 as inductions which may have indeed a high degree of proba- 
 bility, but which have always need of being verified by the 
 accordance between the logical consequences to which they 
 lead, and the phenomena directly observed. 
 
 3. The following principle we assume as being in accord- 
 ance with experiment and observation, viz. a particle which 
 is absolutely at rest will continue so, until some cause, extra- 
 neous to itself, begins to operate so as to put it in motion. 
 This principle asserts that matter at rest has no tendency to 
 put itself in motion, and that any motion or tendency to 
 motion which it may possess, must arise entirely from some 
 external cause. To such causes we give the name of forces, 
 and we give the following definition : — 
 
 Any cause which excites motion in a particle, or which 
 only tends to excite it when its effect is prevented or modified 
 
DEFINITIONS. 3 
 
 by any other cause (or which tends to modify e:!;isting motion), 
 is called _yorce. 
 
 And the line of action of the force is the line in which the 
 particle would begin to moye in consequence of the action of 
 the force, if the particle were at rest and perfectly free. 
 
 When several forces act simultaneously on a free particle 
 or on a system of connected particles, the forces will modify 
 each other's effects : if they are so related that no motion of 
 the particle or system takes place, the forces are said to be in 
 equilibrium. 
 
 That part of Mechanics which treats of the conditions of 
 equilibrium of forces (applied to matter) is called Statics : the 
 other part which treats of the conditions of motion is called 
 Dynamics. The two combined constitute the whole subject. 
 
 4. Forces are brought into action by various causes, and 
 different terms are applied to them in different cases. Thus, 
 for example, if one body press against another, each body is 
 subjected to a force acting at the point of contact, — such force 
 is frequently called pressure ; again, when a body is pulled by 
 means of a string, or pushed by a rod, the force exerted by the 
 string or rod is called tension ; again, experience teaches us 
 that if a body be let free from the hand it will fall to the sur- 
 face of the earth in a certain definite direction, — however often 
 the experiment be tried the result is the same, the body strikes 
 the same spot on the ground in each trial, provided the place 
 from which it is dropped remain the same : — this unvarying 
 effect must result from some cause equally unvarying. 
 
 This cause is assumed to be an affinity which all bodies 
 have for the earth, and is termed the force of attraction. It 
 is found to prevail at all parts of the earth ; and is, in fact, 
 included in the general law of gravitation established by 
 
 1—2 
 
4 INTEODUCTION. 
 
 Newton, viz. that every particle of matter attracts every other 
 particle of matter according to a certain law. The name 
 weight is given to the force which the earth's attraction causes 
 a body at rest to exert downwards. The term gravity is fre- 
 quently used in the same sense statically. 
 
 5. We have a simple example of the simultaneous action 
 of two equal forces when a body rests on a horizontal table, or 
 is supported by the hand. The pressure of the table in the 
 former case, or of the hand in the latter, exactly counter- 
 balances the weight of the body, and is equal to it. 
 
 If a body be suspended freely by a string, the tension of 
 the string, which is the force it exerts on the body, is exactly 
 equal and opposite to the weight of the body. 
 
 6. The question may suggest itself to the student whether 
 the weight of a body remains the same at different times. 
 The answer to this must necessarily depend upon experiment, 
 since we have no means of determining, a priori, whether the 
 attraction of the earth remains the same : but if we can ascer- 
 tain that the mechanical effect of the weight of the body is 
 unvarying (for instance, if it deflect a spring through the same 
 space under precisely similar circumstances), the answer would 
 be in the affirmative. But it would be very difficult to ascer- 
 tain whether the spring were under exactly similar conditions 
 at the different times, and so no reliance could be placed on 
 the result of the experiment. We are able, however, to assert 
 from dynamical considerations that the weight of the same 
 body at the same place of the earth's sm-face is invariable. 
 We may also here state, as a result of experiment, that the 
 weight of a body is not altered by altering its figure. It 
 depends solely upon the volume and material. Thus, for 
 
MASS. 5 
 
 example, a cubic inch of iron requires the same effort to 
 support it, whatever be its form. 
 
 This of course we could not know except from experiment ; 
 for we could easily conceive it to have been otherwise, as, for 
 instance, if the attraction of the earth had been of a kind 
 similar to magnetic attractions which do not influence all 
 substances, and which besides do not exert equal influence 
 over those which are subject to them. 
 
 7. Mass. Common experience makes us acquainted with 
 the fact, that the constitution of all material bodies is not the 
 same. Equal volumes of difierent substances are differently- 
 affected by equal forces applied to them. A cubic inch of 
 wood and a cubic inch of lead require different efforts to 
 support them in the hand. Equal weights of different sub- 
 stances occupy different volumes. We are thus led to con- 
 sider a quality of matter to which the term mass has been 
 given. So long, as the volume and constitution of a given 
 portion of matter remain the same, this quality mass remains 
 the same. The mass of a body has been sometimes defined 
 as the qyMntity of matter in it : but this vague definition does 
 not assist us in forming a distinct conception of it. The 
 notion of muss is one as completely sui generis as those of 
 space, time, weight are so : — and as in these cases, so in that 
 of mass, our principal business must be to establish some 
 mode of measuring or comparing different masses. 
 
 Our only means of measuring mass are derived from 
 dynamical considerations, and we shall have occasion hereafter 
 (in Dynamics) to consider this subject again. For the present, 
 if necessary, the student may assume that the masses of bodies 
 are proportional to their weights at the same place on the 
 earth's surface. 
 
6 INTRODUCTION. 
 
 8. Method of estimating and comparing forces. 
 
 When a force acts on a material point, there are three 
 things necessary to be known in order to render the force 
 perfectly determinate, viz. the intensity of the force, the direc- 
 tion in which it acts, and the position of the point where it is 
 applied, in other words, its point of application. These three 
 things may be called the elements of the force: and when 
 the two latter are assigned, i. e. the point of application and 
 the direction, the line of action becomes determinate, — that is, 
 the line in which the particle would begin to move by the 
 action of this force only, if the particle were perfectly free. 
 
 If two forces be applied in opposite directions to a point 
 which is free and at rest, and constitute an equilibrium, they 
 are said to be equal forces. The notion of the equality of 
 two forces will readily lead to the conception of forces having 
 any proposed ratio to one another : thus if two equal forces 
 be applied in the same direction to the same point, we shall 
 have a double force; if in the same way we combine three 
 equal forces there results a triple force, and so on ; so that, in 
 general, to measure forces we have only to adopt the same 
 method as when we measure or compare any homogeneous 
 quantities: i.e. we must take some known force as unit and 
 then express in numbers the relation which the other forces 
 bear to this unit. 
 
 For example, if F represent the unit of force (the weight 
 of a given body for instance), PF will represent a force the 
 intensity of which is P times that of the unit: or we may 
 speak of a force P simply, in the same sense, — the unit of force 
 being understood. 
 
 9. We have seen that the gravitation of bodies to the 
 earth is unceasing, and, as has been observed, the gravity 
 
EEPEESENTATION OF FOECES. 7 
 
 or weight of the same body is invariable ; so that weight 
 affords a very useful means of estimating all statical forces. 
 The tension of a string may be measured by the weight (the 
 number oi pounds if we please) which it will sustain; the 
 force exerted by a string under constraint may be measured 
 by the weight which will just hold it in its constrained posi- 
 tion ; the force of attraction of a magnet may be measured by 
 the weight it would support : — and so of all statical forces. 
 
 The standard of weight in England is the pound Troy, consisting of 5760 
 grains; and it is stated that a cubic inch of distilled water weighed in air by 
 brass weights at 62° Fahrenheit, the barometer being at 30 inches, weighs 
 252'458 such graim; — the pound Avoirdupois contains 7000 such grains. 
 
 5° Geok. IV. 0. 74. 
 
 10. We proceed to explain how forces may be repre- 
 sented geometrically and algebraically. 
 
 The three things necessary to render a force perfectly 
 determinate are (as we have said) its point of application, the 
 direction in which it acts, and its magnitude or intensity. 
 Now if there be two forces P, Q 
 acting at the points A, G in the 
 directions AB, CD respectively, we 
 may take the lengths of the lines 
 AB, CD such that 
 
 .AB : GD = P: Q. 
 
 Or if we take Q for our unit of force and GD for our unit 
 of length, then the force P will be represented geometrically 
 by the line AB ; for this line is drawn in the direction of the 
 force AP, from the point of applicatian A, and also represents 
 the force in magnitude: the convention in this respect being 
 understood to be tliat the line contains as many units of 
 length as the force contains units of force. 
 
 The student must be careful to observe the order of the 
 
b INTRODUCTION. 
 
 letters which indicate the line ; thus AB expresses that the 
 force acts in direction of the arrow from ^towards B; a 
 force represented by BA would indicate a force of equal 
 magnitude, but acting in thei opposite direction, i.e. from B 
 towards A. 
 
 The force JP would be represented algebraieally by ex- 
 pressing in algebraic symbols, the magnitude and position of 
 the line AB which represents the 
 force geometrically : thus its direc- 
 tion would be assigned by assigning 
 the angle at which it is inclined 
 to a known fixed line Ox in the 
 same plane with AB: its magnitude 
 will be assigned by assigning the 
 numerical value of P, the number 
 
 of units of length ; and the point of application A will be 
 assigned by assigning the position of A with respect to the 
 fixed lines Ox, Oy in the same plane with AB. 
 
 11, This mode of representing forces by lines is of great 
 utility, as we shall see more particularly in the next chapter. 
 We may illustrate it here by 
 
 supposing several forces as P, ^ ■?> ^ ^ 5*- 
 
 Q, R to act simultaneously at 
 
 the point A in the same direction : if they would be separately 
 represented by AB, A G, AD, they will when acting simul- 
 taneously be together represented by a line AU, the length 
 of which is equal to the sum oi. AB + AG + AD. 
 
 If one of the forces as R, acts in a direction opposite to 
 that of the others P, Q, we 
 
 shall have to subtract the line -ȣ ^ b c i>'_ 
 AD from the sum of the others 
 
 -^ — ^ — — >- 
 
TRANSMISSION OP FOECE. 9 
 
 AB, AG, and the three would be represented by a line AU 
 equal in length to AB+AC-AB. This is still the algebraic 
 sum of the lines AB, AC, AD, if lines in one direction from 
 A be considered positive, and lines in the opposite direction 
 negative; and generally if any number of forces act simul- 
 taneously at a point and be affected with the sign + or — as 
 they act in a given direction or the opposite, they will be 
 equivalent to a single force represented by the algebraic sum 
 of the several forces ; and if this sum be affected with a 
 positive sign, the equivalent force will act in the direction 
 which has been considered positive ; and if it be affected with 
 a negative sign, it will act in the opposite direction. 
 
 12. From the definition which has been given of equal 
 forces (in Art. 8), it is obvious that two equal forces applied 
 at a point in opposite directions will be in equilibrium. 
 Further, it will readily be granted 
 
 that two equal and opposite forces < ^ ^ ^^ 
 
 P, Q applied at the extremities of a ^ 
 
 straight rigid rod AB and acting — ^ — • • — ■^— 
 
 in direction of the rod will be in 
 
 equilibrium ; — ^for there is no reason that the rod should move 
 in one direction rather than in another ; — and this result will 
 be true whatever be the length of the rod : from hence we 
 infer that P will balance Q at whatever point of the rod Q 
 be applied: in other words the effect of Q is the same at 
 whatever point of the rod 5, C,... it-be applied, the direction 
 remaining the same. 
 
 These considerations lead us to the following principle, 
 called ihQ principle of the transmission o/"_^rce, which we shall 
 hereafter find to be of great utility. 
 
 The effect of a force on a particle to which it is applied will 
 
10 INTRODUCTION. 
 
 he the same, if we suppose the force applied at any point we 
 please in the line of action, provided the point he rigidly con- 
 nected with the original particle. 
 
 This principle — ^which is the fundamental one of the 
 science of Statics — ^will hold whether we consider the particle 
 as isolated, or as a constituent element of a body of finite size; 
 and we shall find it of great use when we wish to transfer 
 the point of application of a force from one point to another 
 for convenience of calculation. We shall not think it neces- 
 sary in every case where the supposition is required, to state 
 that the system is supposed to be rigidly connected, but in 
 any instance where this is not done the student will under- 
 stand it to be so. 
 
 13. As an illustration of the above principle we may 
 give the following. If a weight be supported by 
 the hand by means of a string, the effort which the U. 
 hand must exert will be the same whatever be the 
 length of the string (the weight of the string being 
 neglected), i. e. whether the force, which the hand 
 exerts, be applied at A, or B, or G, or any point in ' c 
 the line of action of the force. 
 
 Ohs. In this example the student will observe 
 that the connection between the points A, B and the fe 
 weight is not a rigid one, and in general when the 
 force Q (fig. Art. 12), which we transfer from the point C to B, 
 acts as in the upper figure,, i. e. tends to draw towards B, 
 the connection between C and B need not be essentially n^di; 
 but the two points may be otherwise connected, as for instance, 
 by a fine inextensible thread ; when however (as in the lower 
 figure) the force tends to thrust B towards C, the connection 
 must be a rigid one. 
 
TRANSMISSION OP FORCE. 11 
 
 14. We have called the example above an illustration, 
 and not a proof of the principle of Art. (12), for as this prin- 
 ciple has been enunciated with reference to a. particle, and since 
 particles as such cannot be subjected to experiment, it would 
 be vain to look for or expect a direct proof of this, or in fact 
 of any other physical law. The student must be prepared to 
 admit its truth as established by evidence similar to that by 
 which other physical laws are established. 
 
( 12 ) 
 
 CHAPTEE II. 
 
 OF FORCES ACTING IN ONE PLANE. 
 
 15. When a system of forces acting on a particle at rest 
 is not in equilibrium, the particle will begin to move in some 
 definite direction, but a single force might be found of proper 
 intensity which when applied independently to the particle 
 and acting in the same direction would cause the particle to 
 move in exactly the same manner ; such a force is called the 
 resultant of the system of forces ; and the constituent forces 
 of the system, with reference to this resultant, are called 
 components. 
 
 In other words, the single force which is capable of pro- 
 ducing the same effect on a particle or system of particles as 
 would result from the combined action of several other forces, 
 is called their resultant. 
 
 We do not enter into the question what the dynamical 
 effect might be if the system of forces were not in equi- 
 librium — but whatever it may be, the resultant is equivalent 
 to the components. 
 
 When a system of forces acting on a particle or body is 
 in equilibrium, the particle has p 
 
 no tendency to motion, and the j^-, 
 
 resultant is consequently nil. 
 Hence when a system of forces 
 (as P, Q,B,...) is in equilibrium , -^ 
 
 one of them (as P) may be regarded as counterbalancing the 
 combined action of all the rest, Q, B, S. It appears then that 
 the remaining forces ( Q, R, 8) produce the same effect on the 
 particle as would result from a single force equal and opposite 
 
OF FOECES ACTING IN ONE PLANE. 13 
 
 to P. We infer then, that when a system of forces acting on 
 a body is in equilibrium, any one of the forces is equal and 
 opposite to the resultant of all the rest. 
 
 Again, since the resultant of a system of forces in equi- 
 librium is nil, such a system of forces has no tendency to 
 excite or prevent motion; we may therefore (in any case 
 where we find it convenient) suppose such a system of forces 
 to be annihilated without altering the state of rest or motion 
 of the body upon which they act ; or stating this principle 
 more generally, any system of equilibrated forces may be 
 applied to or withdrawn from a body without affecting its 
 state of rept or motion. The student, however, must bear 
 in mind the observation of Art. (13) whenever this principle 
 is employed in dealing with a system of bodies not in rigid 
 connection. 
 
 16. We now proceed to deduce the rules for the com- 
 position of forces, that is, to find the resultant of two or more 
 forces acting simultaneously; and it will then be easy to 
 ascertain the conditions of equilibrium of a system of forces. 
 
 We shall confine ourselves in the present chapter to the 
 discussion of forces acting in one plane. 
 
 The case of forces acting in the same straight line has 
 been already considered in Art. 1 1. 
 
 When two forces P and Q are applied at the same point 
 A in directions inclined to each other 
 at any angle whatever, it is easy to 
 see that some third force R properly 
 applied at the point A would con- 
 stitute an equilibrium with P and 
 Q : for by virtue of the combined 
 action of P and Q the point A 
 
14 OF FOECES ACTING IN ONE PLANE. 
 
 tends to leave the position in which it is ; but since it could 
 move in one direction only, it follows that if we apply a 
 proper force i2 in a direction contrary to this in which it would 
 move, the point could not move at all, i. e. would be at rest. 
 . The three forces P, Q, B acting on the point A would be in 
 equilibrium, and the force B, is equal and opposite to the 
 resultant of the other two. Two forces then, whose lines of 
 action meet, have a resultant. 
 
 Again, it is obvious that this resultant must lie in the 
 plane which passes through the direc- 
 tions of the two components AP, AQ; \ \/ 
 for no reason can be assigned in favour y(A 
 of this resultant's lying in any proposed / / \ 
 position above the plane PAQ, which / \ \ 
 would not hold with equal validity in Y I \ 
 favour of the resultant's being in a per- -P -R <? 
 fectly symmetrical position helow the 
 same plane. 
 
 Further, the resultant must lie within the interior angle 
 PA Q (< 180°) contained, by the directions of the two forces, 
 for it is clear that the point A could not by the action of the 
 forces P, Q move in the plane PA Q, on the side of ^ ^ remote 
 from P and towards D ; and similarly, it could not move on 
 the side of AP remote from Q and towards B : consequently 
 it could only move within the angle PA Q, the direction there- 
 fore of the resultant B must lie within this angle. 
 
 17. There is one case in which we can see h priori what 
 will be the direction of the resultant: viz. when the two 
 forces P, Q are equal ; it is clear in that case that the direc- 
 tion of the resultant bisects the angle between the direction 
 of the two component forces P, Q: for there is no reason 
 
OP FORCES ACTING IN ONE PLANE. 15 
 
 why the resultant should make with one of the component 
 forces an angle different from that which it makes with the 
 other, 
 
 Ois. The student may remark that the conclusion of the 
 preceding article is based on reasoning ex absurdo : instances 
 will have come under his notice, in which the elementary 
 theorems of a subject do not admit of a direct demonstration, 
 but he will regard the proof as equally valid though the 
 demonstration is indirect. The general principle involving- 
 all such proofs as this : If under assigned circumstances, one 
 issue or conclusion and one only can result, and the arguments 
 in favour of two hypothetical issues or conclusions A and B 
 are of equal value, then that hypothetical issue must be the 
 true one in which the two hypotheses A and B coalesce. 
 
 18. We proceed to establish an important theorem which 
 enables us to determine the resultant of any two forces acting 
 at a point : the theorem is called the parallelogram of forces, 
 and may be thus enunciated. 
 
 If two forces acting at a point he represented in magnitude 
 and direction hy two straight lines drawn from that point, and 
 if a parallelogram he constructed having these two lines for 
 adjacent sides, then that diagonal of the parallelogram which 
 passes through the point of application of the forces will 
 represent their resultant in magnitude and direction. 
 
 That is, if the two forces P, Q be 
 represented by AB, AC, and the "^^ * ^- ^-^ 
 parallelogram BO be completed, 
 their resultant R will be represented 
 by the diagonal AD, The same 
 is true if P, Q act at points E, F, 
 provided their directions meet in 
 
16 OF FOKCES ACTIXa IN ONE PLANE. 
 
 some point A. We shall divide the proof of this proposition 
 into two parts ; and 
 
 (i.) To prove that the resultant acts in direction of the 
 diagonal, the forces being commensurable. 
 
 We have seen (Art. 17) that when the forces are equal 
 (AB=A C), their resultant bisects the angle between the direc- 
 tions of the forces, and therefore acts along the diagonal AD ; 
 that is, this first part of the proposition is true for two equal 
 forces. 
 
 Let us assume (a) for the present that it is also true for two 
 
 sets of forces P and Q, P and R — 
 
 . J Q n 1? TT 
 
 equal or unequal; — we can then t'-:;..... * 
 
 prove that it is true for the forces P i "x^ \ "\ 
 
 and Q + R. 
 
 Let P act at A in direction AB, \ '\^^^^' 
 
 Q and R in direction A GE, and let ^ 
 
 AB, A G respresent P, Q in magnitude ; and since R may be 
 supposed to act at any point in the line' AGE which is 
 rigidly connected with A, let R act at G, and let CE repre- 
 sent R. Complete the parallelograms BG, BE. 
 
 Then since by the hypothesis (a) the resultant T oi P, Q 
 acts along AD, let them be replaced by their resultant, and 
 let this resultant be applied at D — which may be done with- 
 out altering its effect (Art. 12). 
 
 Now this resultant T acting at D may be decomposed into 
 two forces P^ , Q^ (equal respectively to P, Q) acting at D in 
 directions GD, D G which are parallel to AB, A G. 
 
 Let T be replaced by P^, Q^, and let the point of appli- 
 cation of P, be removed to C and that of Q^ to G. 
 
 Again P^ and R acting . at G have a resultant acting in 
 direction GG ; let them be replaced by this resultant, and let 
 its point of application be transferred to G. 
 
PAEALLELOGEAM OF FORCES. 17 
 
 {The student may suppose all the points A, 0, D, G, 
 rigidly connected together, Art. (12).} 
 
 We have thus shewn (on the hypothesis a) that the forces 
 P, Q, R which are applied at A, may be supposed to be applied 
 at O without altering their combined effect, — that is, A Q must 
 be the direction of the resultant of P and Q + B m any case 
 in which the hypothesis (a) holds true. 
 
 But this hypothesis is true when P, Q, B are each equal 
 to any the same force f, therefore the conclusion is true for 
 two forces /and 2f, and again, (making Q = 2f,R=f, P=f), 
 it is true for /and 3/ and so by induction it is true for / and 
 mf. Again, putting P = mf, Q = R =f, our conclusion is true 
 for two forces mf, and 2/ and again for mf, and 3/ and gene- 
 rally for »M/and nf: — if to, n be any integers whatever. 
 
 Now any two commensurable forces may, by assigning a 
 proper value to/ be expressed by mf, nf. 
 
 Hence proposition (i) is proved. 
 
 19. (ii.) To prove that the resultant acts in direction of 
 the diagonal, if the forces are incommensurable. 
 
 Let AB, AG represent two such forces. Complete the 
 parallelogram BG, and if AB be not 
 the direction of the resultant, let it 
 be some other line, {AV suppose). 
 Let AG he. divided into an integral 
 number of equal parts each less 
 than BV, — ^which is always pos- 
 sible ; and mark off from GB portions equal to these, — the 
 last division S clearly falling between B and V. Complete 
 the parallelogram GF, — then the resultant of AG, AF mil 
 be in direction AE, and we may suppose this resultant to 
 be substituted for them. 
 
 P.M. 2 
 
18 
 
 OF FOECES ACTING IN ONE PLANE. 
 
 The resultant then oi AG and AB is equivalent to the 
 resultant of some' force in direction AU, together with FB 
 which acts along AB: and this resultant must lie within the 
 angle BAE. But by hypothesis it acts in direction AV, 
 without the same angle, — which is absurd. 
 
 In like manner it may be shewn that no direction but AD 
 can be that of the resultant of the forces AB, AG. The 
 theorem is therefore completely proved so far as the direction 
 of the resultant is concerned: it will be easy now to prove 
 that 
 
 20. (iii.) The diagonal represents the magnitude of the 
 resultant. 
 
 Let AB, AG he, the directions of the given forces, AD 
 that of their resultant : in DA pro- 
 duced take AE of such a length as to 
 represent the magnitude of the result- 
 ant. Then the forces represented by 
 AB, AG, AE balance each other. 
 Complete the parallelograms BE, BG: 
 then AF will be the direction of the 
 resultant of AB, AE, and therefore 
 since each of the three forces AB, AG, 
 AE is equal and opposite to the re- 
 sultant of the other two, — AG, AF are in one straight line. 
 Hence FD is a parallelogram, and .•. AE=FB = AD- i.e. 
 the resultant of AB, AG is represented in inagnitude as well 
 as in direction by AD the diagonal of the parallelogram. 
 
 21. The theorem which we have just proved is of so 
 much importance that it may fairly be considered the fun- 
 damental proposition of Statics. It was enunciated in its 
 present form by Sir Isaac Newton, and Varignon the celebrated 
 
TRIANGLE OP POECES. 19 
 
 mathematician, in the year 1687, — ^probably independently 
 of each other : since that time various proofs of it have been 
 given by different mathematicians, several of which have been 
 reviewed by Jacobi. 
 
 The proof given above is due to M. Duchayla. 
 
 Several very interesting theorems can be readily deduced 
 from the parallelogram of forces : the first we shall give, 
 called the triangle of forces, was announced in the year 1586 
 by Stevinus of Bruges, without any strict proof of it. 
 
 22. The Triangle of Forces. If three forces acting at a 
 point be represented in magnitude and directionhj the sides 
 of a triangle taken in order, they will be in equilibrium. 
 
 Let ABC be the triangle whose sides taken in order re- 
 present in direction and magnitude „ 
 three forces applied at any point, 
 {A suppose). 
 
 Complete the parallelogram BD. 
 
 Then the forces AB, BO applied 
 at A are expressed by AB, AD — (since AD is equal and 
 parallel to BC). 
 
 But the resultant of AB, AD is a force represented 
 hj AG. 
 
 Therefore the three forces represented by AB, BO, CA, all 
 applied at A, are equivalent to AO, GA, which will clearly 
 balance one another. 
 
 Therefore the three forces represented by AB, BO, CA, 
 applied at any point A, will be in equilibrium. 
 
 The converse of this is also true, viz. If three forces acting 
 at a point balance one another, and any triangle be constructed 
 
 2—2 
 
20 
 
 OP FORCES ACTING IN ONE PLANE. 
 
 -7 
 
 x/ 
 
 having its sides parallel to the directions of the forces, the 
 sides of the triangle will be proportional to the forces. 
 
 Let P, Q, R be three such forces 
 acting at any point A, and let AB, 
 AD, represent P, Q, then will the 
 diagonal GA of the parallelogram 
 BD represent B. 
 
 And if ABC be any triangle 
 whose sides are parallel to the sides 
 of ABC, we shall have by similar 
 triangles : 
 
 A'B'-.B'C: C'A' = AB: 
 = P: 
 
 Q 
 
 BC: 
 B. 
 
 GA 
 
 23. From the parallelogram of forces we can easily de- 
 duce the following theorem first stated by Lami, in 1687. 
 
 J^ three forces acting at a point are in equilibrium, each 
 force is proportional to the sine of the angle contained between 
 the directions of the remaining two. 
 
 For referring to the fig. of Art. (20), if P, Q, R, the three 
 forces, be represented by AB, A G, AE, and the parallelogram 
 ^Cbe completed, since AE=AI), we have 
 
 P: Q : R = AB : AG : AD= GD : AG : DA 
 
 = sin DA G : sin AD GisinA GD. 
 Eut sin DAG= sin QAR, 
 
 sin ADG= sin DAB = sin PAR, 
 smAGD=8mDCQ = sinPAQ; 
 
 .: P: Q: R = sin QAR -.sin PAR -.sin PA Q (a); 
 
 or, we may express these relations in the form 
 
 P Q R 
 
 sin ( Q, R) sin {R, P) sin (P, Q) ' 
 
POLYGON OF FOECES. 21 
 
 {Q, B) meaning the angle (< 180°) included between the 
 ■directions of Q and B, — and so of {B, P), (P, Q). 
 
 We can readily obtain the equivalent formulae 
 1^= ^+B!'+2QB(ios {Q, B) 
 Qt= B^ + 1"+ 2BP cos {B, P) 
 ii;'=P^+«'+2Pecos(P, Q), 
 
 by which any one of the three forces P, Q, B is expressed in 
 terms of the other two and the angle between them. 
 
 24. The student will observe that this result is still true 
 if the direction of any one of the forces be exactly reversed ; 
 for example, it would hold if we took a force B' (= B) repre- 
 sented by AD instead of AU, for we should then have 
 
 P: Q: B'= sin B'AQ : sin PAR : sin PAQ, 
 
 but the three forces P, Q, B', would not be in equilibrium ; in 
 fact, the resultant of P, Q being B', the resultant of the three 
 would be 2B'. 
 
 Hence the converse of the theorem of this Article is not 
 true without some additional condition, — such as that each 
 force lies without the angle (< tt) formed by the other two. 
 
 The student however will jiave no difficulty in proving 
 the following : 
 
 If each of three forces acting at a point be proportional 
 to the sine of the angle between the directions of the other 
 two, either the three forces are in equilibrium, or they have a 
 resultant double of some one of the forces. 
 
 25. We may now give a theorem which is an extension 
 of that contained in Art. 22, and is called the Polygon of 
 Forces. 
 
22 OP FORCES ACTING IN ONE PLANE. 
 
 Polygon of Forces. If any number of forces acting at a 
 point he represented in magnitude and direction by the sides 
 of a polygon taken in order, they will be in equilibrium. 
 
 If the forces be represented in magnitude and direction by 
 the sides of the polygon ABODE, ^ 
 
 joining AG, AD we see that forces ^^^-^^s.,,^^ 
 
 represented by AB, BO acting at A A^^:::!^- -^ 
 
 are equivalent to a force AO, which \ '"^^ / 
 
 may therefore replace them. Again, \ "^^^ / 
 
 AG, CD acting at A are equivalent L liJ 
 
 to AD; i. e. AD is equivalent to 
 
 forces AB, BC, CD all acting at A. Again, AD, DE are 
 equivalent to AE; and therefore AD, DE, EA will balance, 
 (Art. 22.) 
 
 Hence the forces represented by AB, BC, CD, DE, EA 
 will be in equilibrium, q.e.d. 
 
 The same mode of proof will hold whatever be the num- 
 ber of the forces, and the student will observe that there is 
 no necessity for all the forces to be in the same plane : — the 
 polygon whose sides represent the forces may have re-enter- 
 ing angles, or some of its sides may intersect each other. 
 The only condition is that the polygon must be a closed one. 
 
 By drawing a line parallel to one of the sides of the 
 polygon, as BC, we might form a new polygon whose sides 
 are parallel to those of the former, but the sides of the two 
 polygons are not in the same pi-oportion. 
 
 Hence the converse of the proposition of this Article is 
 not necessarily true. 
 
 26. 
 
 CoK. From the proposition of the previous Article 
 
RESOLUTION OP A FORCE. 23 
 
 we can obtain at once a theorem given loj Leibnitz for deter- 
 mining geometrically the resultant of any number of forces 
 acting at a point. 
 
 From any point A draw a straight line AB to represent 
 one of the forces in magnitude and direction; from the ex- 
 tremity B draw BC to represent the next force; from G draw 
 CD to represent the third force, and so on; and let E be the 
 extremity of the line representing the last force. 
 
 Then if E coincide with A the resultant is nil, and the 
 forces are in equilibrium; but if not, AE will represent the 
 resultant in magnitude and direction. The student will easily 
 deduce this from the preceding Article. 
 
 27. We have seen in Article (18), that if BC be a paral- 
 lelogram, the two forces AB, AG acting 
 at a point A are equivalent to a single force 
 AD acting at the same point; which single 
 force might be substituted for the two com- 
 ponent forces: vice versa if a line AD 
 represent a force, and ani/ parallelogram as 5 C be constructed 
 having AD for a diagonal, the single force AD may be 
 replaced by two forces represented by AB, A C, i. e. AD may 
 be resolved into two forces AB, AC. 
 
 Also, since the number of parallelograms which can be 
 constructed with AD as diagonal is unlimited, it follows that 
 a single force can be resolved into two others equivalent to it 
 in an unlimited number of ways. 
 
 Further, each of the forces AB, A C, may be resolved into 
 two others, in a way similar to that by which AD was re- 
 solved into two, and so on to any extent; so that we arrive 
 ;it the conclusion that a single force may be resolved into any 
 
24 OP FORCES ACTING IN ONE PLANE. 
 
 number of forces we please, the combined action of which is 
 equivalent to the original force. 
 
 On compa-ing the sides of the triangle ADB in which 
 BD =ul(7 we readily obtain 
 
 , sin CAD 
 
 AB=AD 
 
 AO^BD^AD 
 
 sin BAG' 
 
 sin BAD 
 
 sin BAG' 
 
 or if AD represents a force B, we conclude that a force B 
 acting in a direction AD is equivalent to the two forces 
 
 Ji ^^" ^^^f^ in direction AB, =P (suppose), 
 sm BA G 
 
 i?^-H^ AC, = Q. 
 
 sm BA C 
 
 Hence if we put BA G= a, BAD = /3, GAD = 7, we have 
 
 P=b'^, Q = b'^; 
 sm a sm a 
 
 and B' = P^+ (^+ IPQ cos a, 
 
 formulae which enable us to find the resultant of two forces, 
 or to resolve a single force into two others. 
 
 N.B. We shall hereafter meet with instances of the 
 resolution of one force into two others equivalent to it; per- 
 haps the most frequent case which occurs, is when the angle 
 GAB =90°, or the parallelogram becomes a rectangle : in this 
 case a force P acting in direction AD is equivalent to the two 
 forces 
 
 Pcos DAB in direction AB] 
 and PcosDAG AG}' 
 
.KESULTANT OF SEVEEAL FOECES. 25 
 
 28. We may now proceed 
 
 To find the resultant of any number of forces acting in one 
 plane at a point. 
 
 We may proceed geometrically thus, 
 
 Let AB, AG, AD.... represent the forces P, F, P'... 
 Take any two forces AB, AG, 
 complete the parallelogram BG; 
 and A Q, the resultant of P, P', 
 may he substituted for them. 
 
 Find the resultant oi AQ and 
 AD (or P") in a similar way; 
 then the three forces P, P', P", 
 are equivalent to AR, and so on, 
 till the resultant of all the forces is obtained. 
 
 Or we may proceed thus by the 
 aid of trigonometry. 
 
 Through the point draw two lines 
 Ax, Ay at right angles to each other 
 in the plane of the forces, and let 
 the directions of P, P',... make an- 
 gles a, a',... with Ax. 
 
 Then since P, P',... are equiva- 
 lent to 
 
 Pcos a in direction of Ax, and Psin a in direction of Ay, 
 
 P'cosa' P'sina' 
 
 all the forces P, F',... are equivalent to 
 
 Pcos a + P' cos a'+ ... in direction of Ax, 
 
 Psin a +P' sin a' + Ay. 
 
26 OP FOKCES ACTING IN ONE PLANE. 
 
 Or as we may write it % (Pcos a) in direction of Ax, 
 2(Psina) At/. 
 
 If then B be the resultant making an angle 6 with Ax^ 
 we must have ^ cos = 2 (Pcos a), 
 
 ^ sin 5 = 2 (P sin a). 
 
 Whence i2^= {t (P cos a)Y+ {% (Psin a)}'' (i), 
 
 „ 2 (Psin a) .. 
 
 tan5 = =r7n f (n); 
 
 2 (Pcos a) ^ ' 
 
 the results (i) and (ii) determine the magnitude and direction 
 of the resultant. 
 
 COE. 1. If the separate forces P, P'... be resolved in 
 direction of their resultant B and perpendicular to it, the 
 algebraic sum of the former resolved parts will be = B, and of 
 the latter will be = 0. 
 
 CoE. 2. If such a system of forces as is considered in this 
 Article is in equilibrium, the resultant must be zero; i.e. B=0; 
 
 and therefore {2 (Pcos «)}'+ {2 (Psin a)j''= 0, 
 
 which requires that 
 
 2 (Pcos a) = 0, and 2 (Psin a) = 0, 
 
 that is, the sum of the forces resolved in any two directions at 
 right angles to each other must be severally zei-o. 
 
 29. To find the resultant of two forces whose directions are 
 parallel. 
 
 Let A, B be any two points in the lines of action of the two 
 
TWO PARALLEL FORCES. 27 
 
 forces P, Q, which act in the pa- 
 rallel directions AP, BQ. 
 
 At A apply any force S in di- 
 rection BAS, and at B apply an 
 
 equal force S' in the direction ^■ <\J^ lo ^i^^*^' 
 
 ABS', — this will not modify n/ 
 the combined action of the other r 
 forces. 
 
 Now 8, P acting at A, are equivalent to a single force -B, 
 acting in some direction AR^. 
 
 And S', Q acting at B, are equivalent to a single force ^„ 
 acting in some direction BR„. 
 
 Let these two pairs of forces be replaced by R^, i?,,, whose 
 directions will meet in some point ; let the points of appli- 
 cation of ^,, R^^ be transferred to 0. 
 
 Draw OCR parallel to AP or BQ, and 80S' parallel 
 to AB. 
 
 Now let B^ acting at be resolved into two components 
 in directions 08 and 00, which will clearly be (Sf and P, and 
 let R^i acting at be resolved into two components in direc- 
 tions 08' and 00, which will be 8' and Q. 
 
 Then 8 and ;8" being equal in magnitude and opposite in 
 direction, will balance each other, and may therefore be 
 removed, and there remain P and Q acting at in the 
 line OCR. 
 
 Hence, if B be the resultant of P and Q, 
 R = P+Q (i). 
 
 Again, in the triangle A CO, the sides are proportional to 
 8, P, R,, and in the triangle BCO the sides are proportional 
 to 8', Q, R„. Hence 
 
28 
 
 OP FOECES ACTING IN ONE PLANE. 
 
 P OG .8' BG 
 
 S~AG' """ Q 
 therefore multiplying together, and remembering that 8= 8', 
 we get 
 
 P 
 
 Q" 
 
 BG 
 
 'AG 
 
 Hence the point G in the line AB, through which the 
 resultant acts parallel to each of the forces, divides the line 
 AB into segments which are inversely proportional to the 
 forces. 
 
 (i) and (ii) determine the resultant completely. 
 
 30. If the two forces act in opposite directions the method 
 is very similar : the point G lies 
 outside AB; 
 
 audi R = P-Q (iii), 
 
 F 
 'Q" 
 
 BG 
 
 'AG 
 
 (iv). 
 
 It will be observed that (iii) 
 and (iv) are the same as (i) and 
 (ii), if the sign of Q be changed, 
 so that algebraically (i) and (ii) comprise both cases. 
 
 Cor. 1. The position of the point G does not depend 
 upon the direction of the forces. Hence if the directions of 
 the forces be turned through any the same angles in the same 
 direction about the points A, B, the position of G will not be 
 changed. 
 
 COE. 2. In the case of Art. 30, we have from (iv), 
 P 
 
 q" 
 
 BG AB 
 
 ' AG~ AG- 
 
MOMENT OF FORCES. 29 
 
 A TO 
 
 If nowP= Q, -we get -771 = 0, or AC=(jo, and E = 0. 
 
 A system of two equal forces acting in opposite directions 
 and not at the same point is called a couple; — and tlie 
 results iZ = 0, and AO = cis with reference to such a system 
 indicate that a couple cannot be replaced by any single finite 
 force acting at a finite distance. 
 
 31. Moment of a Force. 
 
 The product of a force into the perpendicular distance of 
 its line of action from a given point is called the moment of the 
 force with, respect to the point, or the moment of the force about 
 the point. 
 
 If an axis be drawn through the point at right angles to 
 the plane which contains the point and the direction of the 
 force, this product is called the moment of the force about 
 the a^xis. 
 
 Further, The moment of a force about any line is defined to 
 be the product of the resolved part of the force perpendicular 
 to the line into the perpendicular distance between the line 
 and the line of action of the force. — This perpendicular dis- 
 tance is the shortest distance between the two lines. 
 
 The student will be careful to observe that the force and 
 distance here spoken of are ex- 
 pressed numerically in terms of 
 their respective units; and the 
 moment consequently is the pro- 
 duct of two numerical quantities. 
 Thus if AB represent a force P, 
 and be any "^ given point, — 
 
30 OP FORCES ACTING IN ONE PLANE. 
 
 Op perpendicular to AB, and if m, n be the number of linear 
 units 'ra.AB, Op respectively, then will «i« be the moment of P 
 about 0, or about an axis through perpendicular to the plane 
 ABO. Also since the area of the triangle ABO = \AB. Op, 
 it is obvious that mn = twice the number of units of area in 
 the triangle ABO. We may then represent moments geo- 
 metrically by areas, and the moment of P about would thus 
 be represented by twice the triangle ABO : the unit of moment 
 (i.e. the product of a unit of force into a unit of distance) 
 being represented geometrically by a unit of area. 
 
 Further, the force P would tend to twist the body on 
 which it acts in one direction or the reverse, according as 
 is on one side of AB or the other. We shall for convenience 
 consider the moment of a force negative or positive, according 
 as it tends to twist the body in the same direction as the 
 hands of a watch revolve, or the contrary. 
 
 If P, Q be two equal forces acting in parallel but oppo- 
 site directions — constituting a couple — if G be any point in 
 the plane of the forces, and GBA be perpendicular to their 
 lines of action (fig. Art. 30), we have the moment of the two 
 forces about = P.AG- Q .BC = P. AB= constant, i.e. 
 the moment of a couple is the same about any point in the 
 plane of the couple. 
 
 32. The following proposition is important. 
 
 The algebraic sum of the moments of two forces acting in 
 one plane about any point in the plane is equal to the moment 
 of their resultant. 
 
 When the forces are not parallel it admits of a simple 
 geometrical proof. 
 
MOMENT OF FORCES. 31 
 
 Let AB, A G represent the two 
 forces P, Q, and complete the pa- 
 rallelogram BO, and through the 
 point draw rOs parallel io AG; 
 then taking those moments to be 
 positive which tend to twist a 
 body in a direction opposite to 
 that of the hands of a watch, 
 
 sum of moments of P and Q about 
 =2AAOG-2AABO 
 = parallelogram Gr — 2A.AB0 
 = parallelogram 5 C — parallelogram rZ) — 2AAB0 
 = 2 {AABD - ABOJD - AABO) 
 = 2AA0I) 
 = moment of M the resultant of P and Q. 
 
 The above construction will apply if the point lie within 
 the angle BA G, or the vertically op- 
 posite angle. If lie within either 
 of the supplemental angles of BA G, 
 as in fig. 2, draw Ors parallel to 
 A G, then 
 
 sum of moments of P, Q about 
 ■ =2AAOG+2AAOB 
 = parallelogram Gr + '2AA OB 
 = parallelogram OB — parallelogram Bs + 2AA OB 
 ^'2{AABD + AAOB-ABOD)='2AAOD 
 = moment of R the resultant of P and Q. 
 
 33. It remains to prove the proposition for two parallel 
 forces. 
 
32 OF FOiiCES ACTING IN ONE PLANE. 
 
 Let A, B be two points through 
 which the forces P, Q act, G a 
 point in the line AB through which 
 the resultant B passes. 
 
 Take any point and through 
 it draw Ohca at right angles to 
 the directions of the forces; then 
 since the resultant of P, Q passes 
 through G, 
 
 P.AG=Q.BG, 
 and .'. P.ac= Q-ic; 
 when the forces P, Q act in the 
 same directions (fig. 1), we have 
 sum of the moments of P, Q about the point 
 
 = Q.0b+.P.0a 
 
 = Q{Oc-bc)+P{Oc+acj 
 
 = {Q + P) Oc V Q.bo = P.ac 
 
 = B.Oc 
 
 = moment of the resultant about 0. Q.E.D. 
 
 If the forces act in opposite directions (fig. 2), the student 
 will have little difficulty in proving that 
 
 Q. Oc-P. Oa={Q-P).Oh, 
 
 which expresses the same proposition in this case. 
 
 Obs. The point has been taken in such a position that 
 the moment of the resultant is in each case positive. The 
 proposition is readily proved for any other position of 0. 
 
 COE. 1. If the point be taken anywhere in the line of 
 action of the resultant B, the moment of B vanishes, and 
 
MOMENT OF FORCES. 33 
 
 we conclude that the moments of two fdttea about any point m4 
 the line of action of their resultant are equal in magnitude d/nd 
 opposite in directioh. 
 
 ■ COE. 2. We can readily extend the ptDptiSitiOn of 
 Art. (32) to any number of forces in one plane. For since the 
 sum of the moments of two forces is equal to the moment of 
 their resultant, we may substitute the resultant for the two 
 forces ; we may now combine this resultant with a third, and 
 suppose them replaced by their resultant, and so on whatever 
 be the number of forces. Hence 
 
 The moment of the resultant of any number of forces in one 
 plane, taken with respect to any point in that plane, is equal 
 to the algebraic sum of the moments of the several forces with 
 respect to the same point. 
 
 When the moment of the resultant vanishes, we conclude 
 either that the resultant is nil, or that the resultant passes 
 through the point with respect to which the moments are 
 taken. 
 
 34. The sum of the moments of two parallel forces about 
 any line at right angles to their direction, is equal to the moment 
 of their resultant about the same line. 
 
 Let Opq be any line in the plane of the paper — and let 
 R be the resultant of two 
 
 parallel forces P, Q acting ^^-"'''^ 
 perpendicular to this plane, P,,^''^^ | 
 — their directions meeting ^^,^--'1 j i 
 it in the points E, P, Q. --=^— j, — ^; ^ — 
 
 Draw Pp, Br, Qq per- 
 pendicular to Opq — then if the line Opq is parallel to PQ, 
 these perpendiculars are equal, and — since i? = P+ Q — the 
 P. M. 3 
 
34 OF FORCES ACTING IN ONE PLANE. 
 
 moment of B about jpg is equal to the sum of the moments 
 of P and Q. 
 
 But if Opq is not parallel to PQ, let them meet in — 
 then taking moments about 0, 
 
 B.OR = P,OP+Q.OQ; 
 
 but by similar triangles, 
 
 Br _ Pp _ Qq 
 
 0B~ 0P~ OQ' 
 
 whence we get 
 
 B.Br = P.Pp+Q.Qg, 
 
 which proves the proposition. 
 
 Obs. The proposition may easily be extended to any 
 number of parallel forces. 
 
 35. Let two forces P, Q act in one plane at points 
 A, B oi 2i rigid body, and let be a fixed point of the body 
 about which it might turn freely; 
 if the two forces P, Q balance about 
 0, the force arising from the re- 
 action of the fixed point (which of 
 course passes through the point) 
 must with P, Q constitute a system 
 of forces at equilibrium : in other 
 words, the reaction of the fixed pqint 
 is equal and opposite to the resultant of P and Q. 
 
 lip, q be the perpendiculars from on the lines of action 
 of P and Q, we have, since the moments of P, Q about 
 must be equal and of opposite tendency, P.p= Q .q. 
 
 And the pressure on the fixed point 
 
 = »J(P'+Q' + 2PQ cos PCQ). 
 
PKINCIPLE OP THE LEVEE. 35 
 
 Such a fixed point as is commonly called a fulcrum ; 
 the rigid body, whatever be its form, is called a lever. 
 
 COE. If more than two forces act on the body in one 
 plane, and balance about a fixed point or fulcrum 0, the 
 resultant of the forces must pass through 0, and the algebraic 
 sum of the moments of the forces about must be zero; or in 
 other words the sum of the moments of the forces which tend 
 to turn the body in one direction about 0, must be equal- to 
 the sum of the moments which tend to turn the body in the 
 contrary direction. 
 
 36. Further, any point of a body at rest under the action 
 of any forces may be regarded hypothetically as a fulcrum: 
 for since the body is at rest, no point of it has a tendency to 
 move; we shall not therefore disturb its equilibrium or the 
 relations between the forces by supposing any point we please 
 to be incapable of moving, i.e. by supposing it to be a fixed 
 fulcrum. — Hence the sum of the moments of the forces about any 
 point whatever must be zero,-^if the forces be in equilibrium. 
 
 The principle of this article when applied to a rigid body 
 in equilibrium is frequently referred to as the principle of the 
 lever. 
 
 37. The theorem stated in Article (32) admits of the fol- 
 lowing simple analytical proof, — in the case of forces which are 
 not parallel. 
 
 Let AP, AQ\)Q the directions of two forces P, Q whose 
 resultant B acts in direction AB. Let 
 be any point in the plane PAQ, 
 join AO and draw Op, Oq, Or per- 
 pendiculars to AP, AQ, AB. If the 
 forces P, Q be resolved in direction of 
 AO and at right angles to AO, the 
 
 3—2 
 
36 OF FOKCES ACTING IK ONE PLANE. 
 
 sum of the parts resolved in the latter direction wiU 
 = P. sin FAO + <9 . sin QA 0, 
 
 and E . sin RA is the resolved part of B in the same direc- 
 tion ; hence from the nature of a resultant 
 
 P. sin PJ.0+<9. sin QAO = R.miRAO. 
 Multiply each term of this equation by A 0, then 
 P.AO&mFAO+ Q. AO sin QAO = a. AO sin BAO, 
 or P. Op+Q. Oq = B. Or; 
 
 a result which expresses that the sum of the moments of two 
 forces about any point in the plane in which they act is egyud to 
 the moment of their resultant about the same point. 
 
 38. We are now in a position 
 
 To find the conditions of equilibrium of a system of forces 
 acting in one plane. 
 
 We have seen (Art. 32) that the sum of the moments of 
 a system of forces in one plane about any point is equal to 
 the moment of their resultant. Hence if the sum of the 
 moments of the forces about any proposed point A be zero, 
 either the resultant is zero or the direction of it passes through 
 A. Again, if the sum of the moments about another point B 
 be zero, the resultant, if there be any, must pass through B ; 
 i.e. it must act in the line AB. If, further, the sum of the 
 moments of the forces about a third point G (not lying in the 
 line AB) be also zero, it would follow that the resultant, if 
 any, would act in each of the lines AG and BG, which is 
 absurd. Hence the resultant must be zero, and consequently 
 the system of forces in equilibrium. The conditions of equi- 
 librium then of a system of forces acting in one plane on a 
 rigid body or system are these three: " The sums of the moments 
 
CONDITIONS OF EQUILIBRIUM. 
 
 37 
 
 of the forces taken with respect to three points in the plane {but 
 not lying in one straight line) miist be severally zero." 
 
 Obs. There are then three and only three mechanical 
 conditions for the equilibrium of a systefii of forces acting 
 in one plane. 
 
 39. The conditions of equilibrium obtained In the pre- 
 ceding article may be expressed analytically somewhat dif- 
 ferently as follows : 
 
 Let the system of forces be referred to two lines Ax, Ay, 
 
 A 
 
 \ 
 
 at right angles to one 
 another in the plane of 
 the forces. 
 
 Let be the point 
 of application of any 
 one of the forces P, and 
 let P be resolved into 
 two components X, Y, 
 in directions parallel to Ax, Ay respectively. 
 
 If X, y be co-ordinates of 0, and a, b of any point B, the 
 moment of P about B 
 
 = {x-a)Y-(y-b)X. 
 
 And if similar expressions be taken for each of the 
 forces of the system, 
 the sum of the moments about B 
 
 =^%{{x-a)Y^{y-h)X] (1). 
 
 Similarly, if a', V be co-ordinates of 0, another point, 
 sum of moments about 
 
 =^t{{x-a')Y-(:y-V}X\ (li), 
 
 and sum of moments about A 
 
 =^t{xY-yX) (iii). 
 
38 OF FORCES ACTING IN ONE PLANE. 
 
 Now if A, B, G be tiree points not in a straight line, 
 the conditions of equilihrium are that (i), (ii), (iii), must be 
 severally zero ; 
 
 .'.t{xY-yX) = (i... (iv),, 
 
 t[{x-a)Y-{y-l)X} = Q... (v), 
 
 t{{x-a!)Y-{y-l')X] = ... (vi), 
 
 (iv) and (v) combined, give aSF— &SX= 0, 
 
 .(iv) and (vi) a:tY-h'tX = 0, 
 
 from these latter two we get (since -, is not = TT, — the 
 points A, B, C not being in one line), 
 
 2X=0, tY=0, 
 which with (a), 
 
 t{xY-yX) = 0, 
 
 are the conditions of equilibrium ; 
 or we may interpret (&) as follows : 
 
 In order that the forces may he in equilihrium, the sums 
 of the resolved parts of the forces in tiw directions at right 
 angles to each other must severally he zero, and the sum of the 
 moments ahout some one point must he zero also. 
 
 40. The conditions (a) of the preceding article, might 
 have been obtained directly from (i) by the consideration that 
 if the system is in equilibrium, the resultant is zero, and 
 therefore the sum of the moments about any and every point 
 must = 0, i.e. the expression 
 
 t[{x-a)Y-{:y-h)X] 
 
 must = for any and every value of a, h — ^which can only 
 be satisfied by having each of the conditions of (a) satisfied. 
 
CtENERAL EEMi:EkS. 39 
 
 COE. 1. We may further interpret the equations of con- 
 dition (a) thus,— SZ=0, %Y=0, indicate that the body 
 must have no tendency to move parallel to itself, (i. e. without 
 rotation) in direction of Ax or A^ respectively, and the con- 
 dition "t {xY — yX) = Q indicates that it must have no tend- 
 ency to twist about the point A. That is, there must be no 
 motion of translation or rotation. 
 
 CoE. 2. If the system of forces be not in equilibrium, 
 and «, h be co-ordinates of any point in the line of action of 
 the resultant B, we must have 
 
 2[(a;-a)F-(2,-J)Z} = 0; 
 
 and regarding a, h as current co-ordinates, this will be the 
 equation to the line of action of B, or if we use x', y' instead 
 of a, h in accordance with the usual notation, we may arrange 
 the equation (i) in the form 
 
 x'%Y-y'tX='Z{xY-yX). 
 
 If ^ be the angle which R makes with Ax^ we should 
 easily get i2 cos ^ = S {X) , ^ sin ^ = 2 ( F), 
 and therefore i? = {tXy + {t 7)1 
 
 41. General Bemarhs. 
 
 Before closing this chapter, we may make a few remarks 
 which may be some guide to the student in applying the 
 principles and results of this chapter to the solution of 
 problems. 
 
 The forces which affect a body's state of eqlilllbrlum 
 must arise from some agent external to the body, such as 
 (i) the tension of a string attached to a particular point of 
 the body ;■ (ii) the Action of a rod in contact with the body, 
 and which may be a pulling or a thrusting action ; (III) the 
 pressure arising from some other body in Contact Ti^ith it 
 
40 OF rOECES ACTING IN ONE PLANE. 
 
 either at a point or over' a finite surface ; (iy) the attvaetive 
 or repulsive force exercised by sojne external afeptj and 
 which may be conceived as acting like the tension of a string 
 or the thrusting of a rod. 
 
 42, I. Mutual Pressure of smooth and rough Hwfaces. 
 
 If two bodies be in free contact at one point G, there is a 
 mutual action between them, the direction 
 of which passes through that point, Draw ^ ^ m> 
 the common tangent plane at C. v/ 
 
 Then, (i) if the surfeces be smooth, they a v s 
 can exercise no tangential action on each J j~^ 
 other ; the mutual force between them must iK 
 
 therefore in this case be in the common / ^\ 
 
 normal, and the pressure on each body will ^ 
 
 tend within the body; for instance^ the body 
 A will exert a force on B in direction GB, and vice versa. 
 
 (ii) If the surfaces be rough, the mutual pressure between 
 the surfaces may be resolved into two, P and F, one in the 
 direction of the normal, and the other in the tangent plane ; 
 the latter is counteracted by the tangential force brought into 
 play by the roughness of the surfaces ; each of these com- 
 ponent forces (normal and tangential) which act on one bpdy 
 are severally equal and opposite to the corresponding forces 
 acting on the other body. 
 
 If the full amount of friction which the jo\ighnes9 pf the 
 surfaces can give rise to is brought into ex;ercige, then, as 
 will be seen in Ch^p. III. (to which the student is referred), 
 F=fi,P, -^/^ being some qijiantity found by expeajiiHent ; and 
 the direction in whiqh the friction aeta in the tangent pl»ne is 
 e?;actly opposite tO the direction in which the point G would 
 
TENSION OP STKINGS. 
 
 41 
 
 tend to slide if the surfaces were for an instant supposed 
 smooth ; of course the full amount of force which the rough- 
 ness of the surfaces is capable of exercising will not in every 
 case be brought into action ; no more, in fact, will be exercised 
 than is necessary to prevent a tangential sliding motion, 
 
 II. The same principles apply 
 in the case of a rod in free contact 
 with a smooth or rough surface. 
 
 If a rod be connected by a free 
 compass-joint or hinge with another 
 rod (or with a body), there will be a 
 force exercised on each rod equal in 
 magnitude and opposite in direction. 
 
 If we wish to find the magnitude 
 and direction of this mutual reaction, 
 we must assume some unknown force 
 B acting in an unknown direction, 
 
 and obtain equations for determining them by taking the 
 conditions of equilibrium of each rod. It will not unfre- 
 quently be the case, however, that the symmetry of the parts 
 of the system will enable us to assign at once the direction 
 or magnitude of R, or both. 
 
 43. III. Tension of Strings, 
 
 If we consider a string as a liue of consecutive particles, 
 the force which binds successive particles of the string toge- 
 ther is called the tension, and since each particle of the string 
 is urged in opposite directions by the forces which the con- 
 secutive particles on either side of it exercise upon it, these 
 forces must be equal and opposite j i.e. on each element of 
 the string there are two tensions, equal and opposite. If 
 
42 OP FOEGES ACTING IN ONE PLANE. 
 
 we neglect the weight of the string, the tension at all points 
 of the same rectilinear portion Is the 
 
 same : for If A, B he any two points < ^ ^> 
 
 of the string AB, It Is obvious that 
 
 the tensions at A and B must be equal, otherwise the string 
 
 would move. 
 
 (I) Also the tension of the string Is not altered if it pass ovet 
 a smooth surface ; for let pq be a 
 small element of the string on the 
 smooth surface, — -pq may be re- 
 garded as a small arc of the circle 
 of curvature at the middle point of 
 pq, and we may consider pq as a 
 rigid body kept in equilibrium by 
 the tensions t, i at;^, q acting along 
 
 the tangents pt, qi, and by the reaction of the surface JS, 
 which acts along the line OB bisecting the angle p Oq, — since 
 the arc pq Is symmetrical with respect to OB, and the re- 
 sultant pressure of the surface will therefore act along OB. 
 
 Since then t, t', B are three forces in equilibrium, we get 
 by resolving them perpendicular to OB 
 
 tcospOB = t'co3qOB; 'butqOB=pOB, .•.t = t'; 
 
 i.e. 'the tension at successive elementary distances is the 
 same, and therefore it is so at finite distances. Hence If 
 the string be pulled by forces T, T' at Its two ends, we must 
 have T= T' — tension at any intermediate point. 
 
 A stricter proof of this result will be given hereafter, 
 Art. (65, 66). 
 
 (il) If a string pass over a rough surface, the tension at 
 successive points will not be the same. 
 
TENSION OF STRINGS. 43 
 
 If P, Q be the tensions at the extremities of a string which 
 passes in one plane over a rough curve or surface, and the 
 string be on the point of motion in the direction in which 
 P acts, then P= Qe*"*: where /* = coefficient of friction (see 
 chapter on Friction) and ^ is the circular measure of the 
 angle included between the normals to the curve at the points 
 where the string quits the curve. Art. (66). 
 
 (iii) Elastic Strings. If an elastic string whose natural 
 or unstretched length is I be stretched to a length I by the 
 action of a tension t which is uniform throughout the length of 
 the string, it is found by experiment that the extension 
 V —I is proportional to the natural length I, and also to the 
 tension t, so that 
 
 V — la: U = — say ; 
 
 or I' 
 
 -'('-:). 
 
 where e — which is called the modulus of elasticity of the 
 string — is some quantity depending upon the nature of each 
 particular string. If t = e, then I' = 2l, i. e. if the string be 
 subject to a tension equal to the modulus of elasticity, it will 
 be stretched to twice its natural length. 
 
 N.B. In all the above cases the weight of the string is 
 neglected. 
 
 44. When a system of bodies is at equilibrium under the 
 action of any forces, no part of the system has any tendency 
 to move ; and we shall not affect the statical condition of the 
 system, if we suppose any part or parts of the system to be 
 deprived of the power of motion ; as, for example, by sup- 
 posing a body in contact with others to be rigidly attached 
 to them. In accordance with this principle, which is of 
 
44 
 
 OF FORCES ACTING IN ONE PLANE. 
 
 frequent and useful application, when we are considering the 
 equilibrium of any system, or part of a system of bodies, we 
 may suppose the portion under consideration to be rigid; 
 which supposition will enable us to lay out of account all 
 mutual forces within the system. As an illustration of the 
 application of this principle, sup- 
 pose a system of bodies A, B, 0, D 
 kept at rest under the operation of 
 a known system of forces ; in con- 
 sidering the equilibrium of the 
 body C (for example) we may re- 
 gard the rest A, B, D as rigidly 
 
 connected together, so that we thus avoid the introduction of 
 the mutual pressures between A and D, and B and D. 
 
 Again, if a string passes round a surface B, quitting it at 
 the points V, T, we may suppose the string to be attached to 
 the body B at the points V, T, which is equivalent to sup- 
 posing that the portion of the string in contact with the body 
 is rigid and rigidly attached to the body. 
 
 45, The case of a body kept in equilibrium by three 
 forces acting in one plane, is of so frequent occurrence as to 
 deserve special notice. 
 
 The conditions of equilibrium of a body kept at rest by 
 three forces P, Q, B In one plane may be stated thus : 
 
 I. If their directions are parallel : — 
 
 (I) Their algebraic sum must be zero, or 
 B=-F+Q, Art. (29). 
 
 (II) The moments of any two 
 of the forces about a point in the 
 line of action of the third must be 
 equal and of opposite tendency, or 
 
EQUILIBRIUM OF THBEE FORCES. 45 
 
 P.AG^Q.BO, or P.AB^R.BG, or B.AC=-Q.AB, 
 which are all equivalent to one another, Art* 33, Cor. 1. 
 
 II. If their directions are not parallel: — 
 
 (i) Their lines of action must meet in a point, Art. (22). 
 
 (ii) Each force is proportional to 
 the sine of the angle between the other 
 two, the. direction of each force lying 
 without the angle formed by the other 
 two. Art. (23). 
 
 For this latter condition we may 
 substitute the following; viz. each force 
 is equal and opposite to the resultant 
 of the other two, 
 
 {for example B = ^{F'+g' + 2PQ cos FCQ)}. 
 
 CoE. In each case I. and II. we have three^ and only three 
 conditions from mechanical considerations. In I. the forces 
 are parallel, which with (i) and (ii) constitute the three con- 
 ditions. In case II., (i) gives one condition, and (ii) two in- 
 dependent conditions ; three in all. If in any problem more 
 than three quantities have to be determined, the subsidiary 
 equations of condition must be sought for from geometrical 
 considerations ; and whenever the weight of a body is one of 
 the forces to be taken into account it must always be sup- 
 posed to act in a vertical line passing through the centre of 
 gravi^ of the body. (See Chap. V.) 
 
 If more than three forces act on the body in one plane, 
 the conditions of equilibrium given in Art. (38) or the 
 equivalent forms given in Art. (39), will furnish all the re- 
 quisite mechanical equations. Geometrical relations as stated 
 above must furnish any additional data required. 
 
46 OP FORCES ACTING IN ONK PLANE. 
 
 These considerations -would equally apply in the case of 
 the preceding Article when only three forces act, and may be 
 used by the student instead of them, at his discretion. 
 
 The following problems are worked out as examples : 
 
 46. I. To find, the. conditions of equilibrium of a uniform 
 heavy rod, which is suspended hy two strings attached to its 
 ends, the strings being of given length and attached to the same 
 fixed point. 
 
 If AB be the rod, G its middle point, A G, BC the two 
 strings attached to a fixed point G, 
 we have the rod kept in equilibrium 
 by three forces, viz. the tensions 
 {T, T) of the two strings and the 
 weight of the rod which acts through 
 G'va.2i vertical line. 
 
 Since the two tensions act through 
 G, the third force must also pass through G, and therefore 
 GG must be vertical; this determines geometrically the 
 position of the rod, and if we draw Gp parallel to A 0, the 
 sides of the triangle GGp taken in order, are in the directions 
 of the three forces. 
 
 Hence, T : T : W= Gp : Gp : GG (a). , 
 
 Since the triangle GGp is geometrically determinate, the 
 proportions (a) determine T, T . 
 
 We may express T, T thus analytically. 
 Let ACG = a., BGG = P, then a, jS are known quantities 
 since all the lines of the figure are of known length. 
 
 Then, T: T : T7=sinj8 : sina : sin(a + i8); 
 sin/3 „, _^ sina 
 
 T= W . ° ^^, , T = W-^ 
 
 sin(a + /3)' sin(a+yQ)' 
 
PEOBLEMS. 47 
 
 47. II. Two spheres are supported iy strings attached to 
 a given point, p,nd rest against each other: find the tensions 
 of the strings. 
 
 Let A, B be the two spheres, T, T- the tensions of the two 
 strings, W, W' the weights of the spheres 
 which may be supposed to act through 
 their centres. 
 
 Then in considering the equilibrium 
 of ^, there are three forces acting on it, 
 Tiz. the tension of the string T, the weight 
 W&ni the pressure at the point of contact 
 0: now the directions of the two latter 
 forces pass through A, hence the third 
 does so also ; i. e. the direction of the string passes through 
 A, or CTA is a straight line. 
 
 Similarly, CT'B is a straight line. 
 
 Further, in considering the equilibrium of the whole, we 
 may regard A and B as forming one rigid body. Art. (44) ; 
 let G be the centre of gravity of the two spheres. 
 
 Hence, since the forces which keep the united mass of A 
 and B at rest are T, T' and W-\- W, of which the two former 
 pass through C, and the latter acts in a vertical line through 
 G, this vertical line must pass through G also, or CG must 
 be vertical. 
 
 This determines the position of equilibrium geometrically, 
 and the tensions T and T' might be found as in the last 
 problem : the only diflference being that G is not necessarily 
 the- middle point of ^5. 
 
 If it be required to find the mutual pressure (P) between 
 the two spheres, we have by considering the equilibrium of 
 the three forces which pass through. A, 
 
48 OF FOKCES ACTING IN ONE PLANE. 
 
 P : W= sin TAW : sia TAB 
 :^ sin ACQ -.smTABi 
 a known ratio, since A CG, TAB, are known or easily found. 
 Hence P is determined. 
 
 48. III. A heavy partiele {weight W) is attaehed to the 
 middle point of a rodAB without weight, the ends of which rest 
 against two inclined planes at right angles to one another : the 
 vertical plane which passes through the rod heing at right angles 
 to the line of intersection of the two planes. Find the position of 
 equilibrium of the rod, and the pressure on each plane. 
 
 Let B, B' be the pessurea which the planes exert on the 
 rod at its ends A, B, 
 then the only forces ■^.g/ 
 
 which act on the rod are /'i '■--. 
 
 B, E and W, and there- 
 fore when the rod is in 
 a position of ecLuilibrium 
 these forces must satisfy ) ^.->--'e'. l:!"*^^!/ ."'.'; ........ y\ 
 
 the coEflditions of equi- 
 librium of three forces in one plane. Art. (45), Case II. 
 
 Let the normals to the planes at A, B meet in C, then 
 the vertical line through G must pass through W; and 
 therefore the diagonal CWO of the rectangle GO must be 
 vertical. 
 
 Hence a-6 = ^TBO = ^WOB=^'^ - a, 
 
 ■■■ ^ = 2a-|^ (i). 
 
 Also, 
 
 B:B': W=&inEGW: sinBCW : sin E GB 
 = sina : cos a : 1. 
 
PROBLEMS. 
 
 49 
 
 Since It' CW=7r- a, IiGW='^ + a, SCB=-; 
 
 whence R= TFsina, R' = Wcosa. (ii). 
 
 (i) and (ii) express the complete solution. 
 
 If a < 7 , i. e. if OB be that plane which is least inclined 
 
 to the horizon, 6 assumes a negative value, which indicates 
 that the rod is inclined in the other direction to the horizon. 
 
 49. IV. A rectangular picture-frame is suspended hy a 
 string attached to the ends of one side of the frame, the string 
 passing over a smooth peg ; determine the position of equilihrium. 
 
 Let C be the peg over which the string AGB passes 
 freely ; we may suppose the weight 
 of the frame to act at G, the point 
 where the diagonals intersect, and 
 which is the centre of gravity of the 
 frame. Then the forces which must 
 be in equilibrium are the weight W jj.. 
 which acts in the vertical line through 
 G, and the tensions which act on the 
 frame at ^, 5 in directions AC,BC; 
 hence the vertical line through G must pass through C; 
 i. e. CG must be vertical. Also, since the peg is smooth, the 
 tension of the string is the same throughout its length. 
 
 Since then of the three forces in equilibrium whose direc- 
 tions pass through G, two of them, viz. the tensions at A, B, 
 are equal in magnitude, the direction of the' third GG must 
 bisect the angle A GB. The problem then is reduced to the 
 following geometrical one. 
 
 P. M. 
 
 4 
 
50 
 
 OF FORCES ACTING IN ONE PLANE. 
 
 To determine the position of the string ACB of given 
 length in order that the line CG, 
 passing through G a given point in 
 the frame, may bisect the angle 
 A CB. We may construct it geo- 
 metrically thus, — with A, B as foci _/ 
 describe an ellipse whose major-axis / 
 equals A GB ; also describe a circle \ 
 round the triangle ABG. The " — ' 
 points of intersection of this ellipse 
 and circle ( G and G') will determine 
 the point G of the string, which i) 
 
 must coincide with the peg; for the arcs AG, BG being 
 equal, it is obvious that GG, G' G bisect the angles AGB, 
 AG'B, respectively. — There is a third position of equilibrium, 
 viz. when the string is in the position A G"B, G" being the 
 extremity of the minor axis of the ellipse, — for in this case 
 also C"(? bisects the angle AG" B. 
 
 It appears then, that if the circle and ellipse intersect, 
 there are three positions of equilibrium. But if they do not 
 intersect, G, G' have no existence, and there is only one 
 position of equilibrium. The condition that there may be 
 three positions of equilibrium is that the two curves may 
 intersect; i.e. the length of string must be <2 chord ^T. 
 If AB = a, AD = G, the condition becomes 
 
 l<a 
 
 ^/(a=+c^' 
 
 50. V. The determination of the action of a hinge or 
 joint is well illustrated in the following problem. 
 
 Three rods, forming a triangle, are connected hy free joints 
 or hinges at their extremities, and the system is at equilibrium 
 
PEOBLEMS. 
 
 51 
 
 •when certain firces are appliedperpewdicularly to the rods at 
 their middle points — shew that 
 
 (i) the force applied to any rod is proportional to the length 
 of the rod; 
 
 (ii) the strain at each angular paint is the same, and 
 acts in the direction of a tangent to the circle circumscribing 
 the triangle; 
 
 (iii) this strain is proportional to the radius, of the circle. 
 
 (i) Since the mutual action at any one of the hinges 
 will be equal in magnitude 
 and opposite in direction 
 upon the two rods which 
 meet at that hinge — the 
 strains of the hinges may 
 be left out of considera- 
 tion when we are consi- 
 dering the conditions of / 
 
 equilibrium of the three J;' "c" "ff 
 
 rods as one system. 
 
 Now the directions of the forces P, Q, B meet in the 
 centre of the circumscribing circle ; — and since P, Q, M are in 
 equilibrium we must hare 
 
 P: Q:B = smQOB: sinBOP: sinPOQ 
 
 = smA : sinB : ainG=a : b : c, 
 
 which proves the first part of the problem. 
 
 (ii) Let us consider the conditions of equilibrium of one 
 rod AB, and let 8, T be the strains which the hinges at A, B 
 exert upon the rod AB in directions making angles 77' say, 
 with AB respectively. 
 
 4—2 
 
52 OF FORCES ACTING IN ONE PLANE. 
 
 Since AB is in equilibrium under the action of the three 
 forces *Si, T, B, the directions of these forces must meet in a 
 -point C suppose — and since B bisects AB at right angles we 
 easily infer that 7= G'AB= C'BA=y', and 8=T. Simi- 
 larly the strain at G is equal to 8 or T — therefore the strain 
 at each angular point is the same. 
 
 Also the strains at A, B make equal angles 7 with AB, 
 similarly the strains at ^, C upon the rod AG make equal 
 angles — /S say — with AG; and the strains at B, G equal 
 .angles — a. say — with BG. 
 
 From the geometry we readily see that 
 a-\-l3+G=rr, 
 
 /S + 7 + -4=7r, and A + B+ G = '7r, 
 <y+a + B=Tr, 
 .-. a+;S + 7 =7r, 
 
 and .: a = A, /3 = B, 'y=G; 
 and it follows that the direction of the strain at any hinge is 
 a .tangent to the circumscribing circle. Hence the second 
 part of the proposition is proved. 
 
 (iii) Since S at A, and 8 at B balance B, we have 
 2fi'sin7=^ or 2SsiaG = B, 
 but if r be the radius of the circumscribing circle 
 2?" sin G=c; 
 
 • ^ = :?= • 1 = 9. 
 " r G "a b ' 
 
 Hence the strain at any hinge bears to any of the forces 
 P, Q, B the same ratio which the radius of the circumscribing 
 circle bears to the side to which the force is applied — which is 
 the third part of the problem to be proved. 
 
PROBLEMS. 
 
 53 
 
 51. 
 forces. 
 
 The following is an exercise on the parallelogram of 
 
 VI. Assuming the truth of the parallelogram of forces 
 (Art. 18) for the magnitude of the resultant, prove it also for 
 the direction of the resultant. 
 
 Let three forces P, Q, B acting in one plane at a point 
 A be in equilibrium, and let them be 
 represented by AB, AC, AE respec- 
 tively. 
 
 Complete the parallelogram BG; 
 then by the assumption the diagonal 
 AD represents the resultant of P, Q 
 in magnitude: — and since any one of 
 the three forces P, Q, B is equal in mag- 
 nitude to the resultant of the other two, 
 it follows that AE = AD. 
 
 Complete the parallelogram BE; then AF represents the 
 resultant of P, B in magnitude, and therefore AF=AG. 
 
 Hence BF, AD are equal, since they are each equal 
 to^^, 
 
 and AF, BD are equal, since they are each equal 
 to AO, 
 
 that is, the opposite sides of the quadrilateral AFBD are 
 respectively equal to each, other, and therefore AFBD is a 
 parallelogram. 
 
 Hence AD, AE being each parallel to BF are in the 
 same straight line ; — ^which proves the parallelogram of forces 
 for the direction of the resultant. 
 
( 54 ) 
 
 GHAPTEE III. 
 
 OF TEICTION. 
 
 52. When a heavy tody rests on a plane horizontal 
 surface, on a table for example, and we wish to make it slide 
 along the surface, we encounter a resistance to this motion; 
 there exists between the particles of the body and the table 
 an adhesion which resists their separation, and this adhesion 
 is only overcome by applying to the body a force of traction 
 sufficiently great. This adhesive force is calledijrictt6n,andi' 
 the magnitude of the force which is necessary to overcome 
 the resistance to motion will be a measure of the friction. 
 
 More generally, when one surface presses against another, 
 if the direction of this pressure be not normal to the surfaces 
 in contact, there will be a tendency of one surface to rub or 
 slide ovev the other.; and no sliding motion will ensue, unless 
 the resolved part of the pressure along the surface be sufficient 
 to overcome the friction. When a body is Just on the point 
 of sliding, it is said to be in a state hordering on motion, and 
 the greatest amount of friction which the surfaces can exert 
 is then in operation. In other cases no more friction is called 
 into action than is just sufficient to balance the part of the 
 pressure resolved along the surface in contact. 
 
 In this point of view, friction may be called a self-adjusting 
 force, since it adapts itself to the requirements of each parti- 
 cular case; no more being called into operation than is just 
 necessary to prevent motion. 
 
OF FEICTION. 55 
 
 53. If B be the normal pressure between two surfaces in 
 
 contact, F the friction when the bodies are just on the point 
 
 of sliding over each other, i.e. the maximum friction which 
 
 W 
 the, substances can exercise, the rati© -^ is- called the_ coefficient 
 
 of friction, and is commonly designated by /i : 
 so that F= /xB. 
 
 If in any particular case the full amount of friction which 
 the substances can exert is not called into action, the amount 
 of friction which is actually in operation is one of the unknown 
 forces which it is the object of the problem to determine. 
 
 54. The results of careful experiments made with the 
 object of determining the laws of friction are thus given by 
 Coulomb, and M. Morin : viz. 
 
 (i) When the substances in contact remain the same, the 
 friction varies as the pressure; i.e. /t is the same for the 
 same substances, but will vary for different substances. When 
 the pressure is very great indeed, it is found that the friction 
 is a little less than this law would give. 
 
 (ii) Bo long as the normal pressure between the surfaces in 
 contact remains the same, the whole amount of friction is inde- 
 pendent of the extent of surface in contact. 
 
 These two laws are true when the body is in a state 
 bordering on motion, and also when actually in motion; only 
 it is to be remarked that in the latter case the magnitude of 
 the friction is much less than in the former. If we call the 
 friction in the former case statical, and that in the latter 
 dynamical, we may express the above by saying that the coeffi- 
 cients of dynamical and statical friction are severally constant 
 
56, OF FEICTION. 
 
 for the same substances, Ibut that the dynamical is less than 
 the statical. 
 
 It is also found : 
 
 (iii) That the friction is independent of the velocity when 
 the hody is in motion. 
 
 55. The friction between two bodies will generally be 
 diminished by smearing them with some unctuous substance, 
 as oil, &c., and the friction when they are on the point of 
 moving, or what we may call the friction at starting, is pretty 
 nearly the same as during motion when the bodies are made 
 of hard material, like stone or metal. But in the case of com- 
 pressible substances like wood, the friction at starting is very 
 considerably greater than during motion. When two bodies 
 are placed one upon the other, one of them at least being 
 compressible, the amount of friction at starting will partly 
 depend upon the length of time they have been in contact. 
 For wood sliding upon wood, the maximum friction is attained 
 after a contact of a few minutes ; but for wood upon metal it 
 requires a much longer time, frequently several days for the 
 friction to attain its maximum : but when it has attained 
 this, the friction at starting is not altered by any continued 
 duration of contact. i 
 
 Further, it is found that rolling friction is much less than 
 sliding friction: for example, when a cylinder rolls on a plane, 
 or a cylindrical axis turns within a hollow socket (when there 
 is simply a line in contact and not a finite area), the amount 
 of friction is much less than would be given by the above 
 laws (i) and (ii), for the same amount of pressure. 
 
 The fact that rolling friction is much less than sliding 
 friction is taken advantage of in various contrivances for faci- 
 
OF FEICTION. 
 
 57. 
 
 litating tlie transport of heavy bodies; — thus for instance, 
 heavy blocks of stone or other material are often transported 
 by placing them on a platform beneath which rollers are 
 placed: — the wheels of carriages are examples of the same 
 principle, — the most delicate application of which perhaps is 
 that oi friction wheels, such as those employed in Atwood^s 
 Machine (see Dynamics, Art. 82). 
 
 56. The values of! ytt' for diiferent substances have been 
 determined by experiment, and arranged in tables; the follow- 
 ing may be taken as approximate results in many cases for 
 friction at starting: 
 
 wood upon wood 
 wood upon' metal 
 leather upon wood 
 metal upon metal 
 
 (without oil) /* = "5, 
 
 (with oil) fj,= "2; 
 
 (without oil) /M = '6, 
 
 (with oil) /i = '12; 
 
 (without oil) fi = "63, 
 
 (wetted with water) /i = "87 ; 
 
 (without oil) /A ="18, 
 
 (with oil) ^ = -12; 
 
 When a cylinder of wood rolls upon wood so that there 
 
 is a single line of contact only, /* = tk ; — when the surface in 
 
 contact is a physical point the statical friction is inconsider- 
 able. 
 
 57. To find the coefficient of friction hetween two sub- 
 stances practically. 
 
 Let AB be the plane surface of one substance, upon which 
 
58 
 
 OF FEICTION. 
 
 is placed a mass M 
 of the other sub- 
 stance with its plane 
 face in contact with 
 AB. If the plane 
 AB be horizontal, 
 no friction will be 
 called into action, but if it be gradually inclined more and 
 more to the horizon till the body M is just on the point of 
 sliding down AB, then the full amount of friction between the 
 two substances is called into action and only just prevents M 
 from moving down the plane. 
 
 The forces which act upon M and balance each other are 
 W the weight of M, E the pressure of the plane AB upon M 
 normal to AB, and fiB the friction up the plane AB. 
 
 If (j> be the angle which AB makes with the horizon, 
 the conditions of equilibrium give — resolving along the 
 plane and perpendicular to it, 
 
 W sin ^ = fiB, 
 Wcos<f)= R; 
 
 whence tan <f) = fi. Hence if (f> be observed, the value of fi is 
 known. The angle <j> is commonly called the angle of 
 friction. 
 
 58. The above can only be regarded as an approximate 
 method of determining friction. For a complete account of 
 the refined contrivances which have been employed with this 
 object, the student is referred to the memoirs of Coulomb and 
 M. Morin. 
 
 Experiments have been made on a large scale by Stephen- 
 son, Be Pamhour and others for the purpose of determining 
 
OP FEICTION. 59 
 
 the dynamical friction on railroads — by which, the laws stated 
 in Art. (53) have been substantially confirmed. In some of 
 the more favorable cases detailed by Mr. Nicholas Wood in 
 his Practical Treatise on Railroads the rolling friction from 
 the contact of the wheels with the rails, was about -n^t^ part 
 of the whole weight of the train, — and the friction at the axles 
 about as much more — making the total resistance to the 
 motion of the train arising from friction to be about -g^th part 
 of the load. — But the results of diiFerent experiments differ 
 considerably. 
 
(60) 
 
 CHAPTER IV. 
 
 OF FOECES, THE DIEECTIONS OF WHICH MEET IN A 
 
 POINT — TENSION OP STRINGS ON SMOOTH AND 
 
 ROUGH SURFACES. 
 
 59. We proceed to discuss tlie resultant and the con- 
 ditions of equilibrium of a system of forces whose directions 
 meet in a point, but which do not lie all in one plane. 
 
 60. Theorem. If three forces X, Y, Z, not in one plane, 
 he applied at the same point 
 
 [in space) and he represented hy t\ 
 
 the three lines OA, OB, OC, and 'y"-- 
 
 the parallelepiped OABCE, he ^' 
 
 completed, the resultant E of 
 these three forces will he repre- 
 sented hy the diagonal OR of this T¥ 
 parallelopiped. 
 
 For the two forces X, Y, which are represented by OA, 
 OB two sides of the parallelogram OA GB, are equivalent to 
 a resultant P, which is represented by the diagonal OGoi this 
 parallelogram. 
 
 And since 0(7 is equal and parallel to GR, the figure 
 GR G \.& A parallelogram, and consequently the two forces 
 P and Z represented by OG, 00 sides of this parallelogram, 
 will be equivalent to a resultant R represented by the 
 diagonal OR. 
 
 Hence the resultant of the three forces X, Y, Z, is repre- 
 sented by the diagonal OR of the parallelopiped. 
 
 \R 
 
 OMr^ 
 
 A-^ 
 
 f^-ii^ 
 
OP FORCES, THE DIEECTIONS OF "WHICH MEET IN A POINT. 61 
 
 This theorem is sometimes called the parallehpiped of 
 forces. It is an easy extension of the parallelogram of forces. 
 
 CoE. 1. By the preceding theorem we see how-a given 
 force R may always be resolved into three others, severally 
 parallel to three lines given in space: these three lines not 
 being in one plane, and no two of them parallel. 
 
 For if we take OR to represent the given force R in 
 magnitude and direction and draw through the point 0, 
 lines OA, OB, OC severally parallel to the proposed three 
 lines, we have three planes XO Y, YOZ, ZOX, — and if we 
 draw through the point R three planes severally parallel to 
 these three, the. six planes will form a parallelopiped, three 
 adjacent edges of which OA, OB, OG will represent the three 
 components X, Y,Z. 
 
 CoE. 2. If the parallelopiped be rectangular, we have in 
 the rectangle OAGB, 0G^= 0A^+ Off, 
 
 and in the rectangle OGRG, 0R^= 0G^+ 00", 
 whence OR" = 0A'+ 0B"+ OC; 
 
 and therefore iJ''=X»+r^+^ (i) 
 
 or R = '^{X'+Y" + Z'), 
 the value of the resultant in terms of the three components. 
 
 61. If we wish to express each component in terms of 
 the resultant, and the angles which they make with it, and if 
 we denominate by a, /3, y the angles which the direction of 
 R makes with the directions of X, Y, Z, 
 
 i.e. XOR = oi, Y0R = I3, Z0R = 'y, 
 
 we shall have 0G= OR cosy; 
 and therefore Z=R cos 7 1 
 
 similarly, Y=Rcos^> (ii), 
 
 X = ^ cos a ' 
 
62 OF FORCES, 
 
 comparing (i) with (ii) we get 
 
 cos" a + cos"^ + cos°7 = 1, 
 a well-known relation which holds whenever a, yS, 7 represent 
 the angles which any given line makes with three rectangular 
 axes. 
 
 If we multiply the three equations of (ii) successively hy 
 cos 7, cos /3, cos a, we get by virtue of the relation 
 
 cos'a + cos"/? + cos'7 = 1 ; 
 
 Xcosa+Tcos/3 + Zcos7 = i? (iii), 
 
 which expresses that the resultant is equal to the sum of the 
 resolved parts of the components estimated in its direction — 
 a theorem which is true for any system of forces which 
 admits of a single resultant: for if each force be resolved 
 into two parts, one in direction of the resultant and the other 
 at right angles to it, these latter parts must be in equilibrium 
 themselves, and there remains the sum of the former parts 
 equal to the resultant. 
 
 62. We can now proceed : 
 
 To find the resultant of any number of forces whose direc- 
 tions pass through a point. 
 
 Let be the point through which the directions of all 
 the forces pass, and through draw 
 any three lines OX, OY, OZ 
 mutually at right angles. 
 
 Let P be any one of the forces 
 acting in direction OP, making 
 angles a, /3, 7 with OX, OY, OZ, 
 then P is equivalent to three com- 
 ponents 
 
THE DIRECTIONS OP WHICH MEET IN A POINT. 63 
 
 Pcosa, PcosyS, PC0S7, 
 acting in directions OX, Y, OZ, respectively. 
 
 Similarly, if F te another force, a', /3', 7' the angles its 
 direction makes with OX, OY, OZ, it is equivalent to 
 
 P'cosa', P'cos/3', P'cosy, 
 
 in direction of the same lines; 
 
 and so on whatever be the number of forces. 
 
 The system of forces is equivalent then to three com- 
 ponents X, Y, Z, which are severally equal to 
 
 P cos a + F' cos a +■■ .in direction of OX or % (Pcos a) \ 
 
 Pcos/3 + P'cos/S' + OF.. .2 (Pcos/S)i..(i). 
 
 Pcos7 + P'cos7' + OZ...^ (PC0S7) I 
 
 Now these three components (i) are equivalent to a single 
 
 resultant B making angles \, /*, v with the line OX, Y, OZ, 
 
 provided 
 
 B, cos \ = S (Pcos a), 
 
 jBcos/i=^ (PcosyS), B,Q.mv = % (PCOS7) (ii), 
 
 and remembering that cos°\ + cos'/i + cos^ v = \, 
 
 these give us the magnitude of the resultant, i. e. 
 
 B? = {S (Pcos a) Y + {S (Pcos ^)Y + St (Pcos 7)}^; 
 
 and this being known, the equations of (ii) give X, /i, v, which 
 assign the direction of the resultant. 
 
 This analytical mode of finding the resultant of a system 
 of forces applied at a point is of course equivalent to the 
 geometrical construction of Leibnitz noticed in Art. (26). 
 
 63. If the forces be in equilibrium the resultant is nil. 
 i.e. P = 0; 
 
64 
 
 OF FORCES IN EQUILIBRIUM. 
 
 and .-. {S(Pcos.a)P+{S(Pcos^)f +{S(Pcos7)}'=0,• 
 which requires 
 
 t (Pcos a) = 0, t (Pcos 13) = 0, S (Pcos 7) = 0, 
 
 the three conditions of equilibrium of a system of forces acting 
 through a point. 
 
 That is, the sum of the forces resolved in three directions 
 mutually at right angles must be severally zero. 
 
 Or we may reason thus: 
 
 In considering any system of forces whose directions 
 pass through a point, if they be in equilibrium, we may (as 
 has been observed before) regard any one of the forces as 
 equal and opposite to the resultant of all the rest. Hence, in 
 any case in which we are discussing the conditions of equi- 
 librium of a body or system of bodies acted on by such a 
 system of forces, we may resolve all the forces in a particular 
 direction (any we please) — and perpendicular to the direction 
 so taken: the conditions of equilibrium then will be 
 
 (i) The algebraic sum of the former resolved parts must 
 be zero. 
 
 (ii) The resolved parts acting in a plane perpendicular 
 to the direction taken, must be in equilibrium inter se, — 
 and must satisfy the condition of equilibrium of forces in one 
 plane : and we may apply the principles established in the 
 second chapter in the same way as if these resolved parts 
 had been the only forces acting. 
 
 And it will in general constitute part of the solution of 
 the problem to shew that these resolved parts are in equi- 
 librium. 
 
TENSION OF A STEING. 
 
 65 
 
 64. The remaining articles of this chapter contain de- 
 monstrations of two results relating .to the tension of strings 
 passing over smooth and rough surfaces referred to in 
 Article (43) — and some properties of a funicular polygon. 
 
 They may be omitted by a student whose previous read- 
 ing has not prepared him for the consideration of s7naU 
 quantities. 
 
 65. The tension of a string which passes in one plane 
 over a smooth curve or surface, is the same at every point — 
 the weight of the string being neglected. 
 
 Let ps be any finite length 
 of the string in contact with 
 the curve, the normals to 
 which at^, s include an ^ ^. 
 
 Let ps be divided into n 
 parts, such that the normals 
 at the extremities of consecu- 
 tive parts include the same 
 angle 9 — so that nO = 0. 
 
 pq the first of these parts, the normals at p, g meeting 
 in 0, t^, ij,,...i„^.i the tension of the string at^, q,...s. 
 
 B the measure of the pressure on the curve at p — then 
 we may regard the resultant pressure of the curve on the 
 element pq of the string as equal to {B + /c) .avcpq, acting 
 in some direction r V intermediate to Op, Oq, and making 
 angles a^, ^Sj say with pO, qO, — so that a^ + ^^=0, and k 
 is some small quantity which vanishes in the limit, when 6 is 
 taken smaller and smaller. 
 
 P.M. 5 
 
66 TENSION OP STRINGS 
 
 Considering now the equilibrium of the element pq^ of the 
 string as a rigid hody — ^resolve the forces upon it parallel and 
 perpendicular to r V, and we obtain the equations 
 
 ^1 sin ttj + i!^ sin /3j = {B + K)j)q (i), 
 
 t^cosa^—t^cos 13^ = (ii), 
 
 equation (ii) may be written in the form 
 
 t,-2t,sm''^ = t,-2t,8m'^ (1), 
 
 and if we write down the corresponding equation for each 
 consecutive element of ^s, we shall obtain 
 
 <,-2«,sin=| = i3-2<3sin=| (2), 
 
 t,-2i„sm''^=t,^-2t^^Bm'^ («), 
 
 adding equations (1), (2) ... (n), we obtain 
 
 f,-2(*,sin^| + ^,sin^|+...+«„sin^|) 
 
 = i„,,-2 («,sin^|i + «3sin=|^+...+<^,sin='|) .... (iii). 
 
 Now if T be the greatest of the quantities t^, t^...t„^^, 
 we see that 
 
 t, sin= I + t, sin' %'+... + «„ sin' ^ < nr sin' f 
 
 Similarly 
 
 6' Trf)' 
 
 < WT — < -2^ 
 4 4m 
 
 .,sin'§ + *3sin'f+...+.^,sin'|<^^. 
 
ON SMOOTH AND EOUttH SURFACES. 67 
 
 If now n be increased indefinitely, ^ remaining unchanged, 
 and therefore 6 being indefinitely diminished — the expression 
 
 -—- vanishes, and equation (iii) becomes 
 
 *i = Wi (i'^)) 
 
 i. e. the tension of the string is the same at every point. 
 
 Further, from equation (i) — suppressing the suffixes — 
 
 ^ ' sm a + sm p 
 
 and if p be the radius of curvature at p, 
 
 p=^ in the limit 
 
 -; ^ — -r = ^^ p : = o, ih thc limit, 
 
 sm a + sin p a + p 
 
 in which case k vanishes ; whence 
 
 t = Iip (v), 
 
 a relation which gives the pressure on the curve at any point 
 in terms of the tension and radius of curvature. 
 Hence in the same curve 
 
 E = - CC-. 
 P P 
 
 Obs. For simplicity we have supposed the string to be 
 
 all in one plane — ^the demonstration might without much 
 
 trouble be extended to shew that the tension is the same at 
 
 every point in whatever manner the string passes freely 
 
 along a smooth surface or tube of any form. 
 
 66. A string passes in one plane over a rough curve or 
 surface, the tensions of the extremities being such that the string 
 is on the point of motion — to find the relation between these 
 tensions, the weight of the string being neglected. 
 
 5—2 
 
68 
 
 TENSION OF STEINGS 
 
 Let P, Q be the tensions at the points where the string quits 
 the curve — and suppose 
 it to be on the point of 
 motion in the direction 
 in which P acts — then 
 the friction at every point 
 of the arc will act tan- 
 gentially in the opposite 
 direction. 
 
 Let ps be any finite 
 length of the string, the normals at p, s including an ^ j>, 
 let ps be divided into n parts — the normals at the extremities 
 of successive parts including the same •^ 9, so that n6 = ^, 
 pq the first of these parts, the normals at p, q meeting in 
 0, <j, *2 ••• ^n+i the tension of the string at^, q ...s. 
 
 R the measure of the normal pressure on the curve at p, 
 /jlR that of the friction along the tangent atp. 
 
 Then we may regard the resultant of the normal pressure 
 of the curve on the element pq of the string as equal to 
 {R + k) . arcpq acting in some direction r V intermediate to 
 Op, Oq, and making nn /. a say with Op, — and the resultant 
 friction on the arc pq as equal to ii.(B,-\- ic). arcpq acting in 
 some, direction inclined at an z /3 say to the tangent at p — 
 a, /S baing each < Q, and k k small quantities which vanish 
 in the limit when 6 is taken smaller and smaller. 
 
 Considering now the equilibrium of the element pq of the 
 string as a rigid body — resolve the forces upon it parallel 
 and perpendicular to the tangent at p, and. we obtain the 
 equations 
 
 <,-*2C0s^= (5 + K)p2'.sina + /t(iZ+«')j52'.cosyS...(i), 
 
 <, sin 6' = (iZ + KJ^g- . cos a - yu, (i2 + «') ^2 . sin /S. . . (ii) ; 
 
ON ROUGH SUEPACES, 69 
 
 ■whence 
 
 t^ - t^ cos g _ (-R + k) sin g + /^ (ii! + k) cos ^ 
 ij, sin 6 {B + k) cos a— fJ'{B + k) sin /8 ' 
 
 Now the second member of this equation becomes =/i 
 if 6 and consequently a and /8- be taken indefinitely small, 
 we may therefore write it = /i (1 + \), X, being some quantity 
 which = when 6 = 0, 
 
 or ^' = cos ^ {1 + /i (1 + \) tan 6] 
 
 = ("l - 2 sin^'l) {1 + jti (1+ \) tan 6} 
 
 where \' is some quantity which — when ^ = 0. Hence 
 log «, - log «, = log {1 + /i^ (1 + X')} 
 
 = /.^(l + V)-^H(l + V)r+- 
 
 = fi9 (l + X^) say, \^ vanishing with 6. 
 Similarly, 
 
 log*,-log<, = /i0(l + \) 
 
 logf„-logf,.„ = M^(l+X„) 
 .•. log ij — log «„^j = \i-nQ (1 + Kj), if «i be the mean value 
 
 of Xj, X^-.-Xn, 
 
 If now w be increased indefinitely, ^ remaining unchanged, 
 and therefore 6 being indefinitely diminished, each of the 
 quantities \,\...\n will vanish, and therefore k^ will do so 
 likewise, and our equation becomes 
 
 log«i-log«„4.i = /t^, or «i = €"*«„„, 
 
70 THE FUNICULAR POLYGON. 
 
 which expresses the relation between the tensions at any two 
 points of the curve of contact. 
 
 If i/r be the angle between the normals where the string 
 c[uits the curve, we have 
 
 P= Qei^'l' (iii), 
 
 if p be the radius of curvature at^, we shall obtain from equa- 
 tion (ii) the result 
 
 i = Jip (iv), 
 
 Ohs. The results of Arts. (65, 66) are true whether the 
 string be elastic or inelastic — if it be elastic, we may remark 
 that in Art. (66) every element of the string must be sup- 
 posed to be simultaneously on the point of motion. 
 
 67. The funicular polygon. 
 
 If a series of n weights P^, P^...P^ be suspended by 
 knots at given points 
 of a string (without 
 weight), and the string 
 be attached to two fixed 
 points A, B, it will 
 when in equilibrium 
 form a polygon in a 
 vertical plane, and is 
 called & funicular poly- 
 gon. 
 
 To find the conditions of equilibrium in such a system. 
 
 Let <j, t^, ^g.-.i^jbe the tensions of the successive portions 
 of the string AP^, P^P^, P^P^...P^B, and let a„ a, ... a,^, be 
 the angles which these successive portions make with the 
 horizon. 
 
 We shall have for the equilibrium of P^, P^, P,... in 
 succession the following sets of equations, 
 
THE FUNICULAR POLYGON. 
 
 71 
 
 «j COS iXj = i, cos «j, 
 t cosa„= i„ cos a.. 
 
 «, sin aj - «2 sin Hj = Pj (I), 
 
 «,sma,-«3sina3 = Pj (ii), 
 
 *„cosa„ = «^jCosa^j, «„sina„ 
 
 •^«+iSma^i = P„. 
 
 If a,, fflj... a^j be the lengths of the successive portions 
 of the strings AP^, P^P^... P^B, and a, h the horizontal and 
 vertical distance between A and B, we have from the geome- 
 try of the figure the following equations, 
 
 .{n + 1). 
 
 ttj cos Kj + ffljj COS a^ + . 
 ttj sin ttj + ftjj sin a^ + 
 
 The sets of equations (i), (ii) . . . (w + 1) are sufficient to 
 determine the 2m + 2 quantities t^, tj^...t^^^, a,, a^ ... a^i, and 
 contain implicitly a complete solution of the problem. 
 
 From the first column of equations in (i), (ii) ...(w) it 
 appears that the tension of each portion of the string resolved 
 horizontally is of the same magnitude, and if we put this 
 = c, so that c = t^cosa^==t^cosa^= .., we obtain from the 
 sets of equations (i), (ii) ... (n) the following results, 
 
 tana. 
 
 tana. 
 
 ■ tan a. = —^ 
 
 ^ c 
 
 ■ tan a, = — ^ 
 
 ^ c 
 
 tan a„ ■ 
 
 •tan 0^1 = : 
 
 •(A), 
 
 from which we may obtain the following relations connecting 
 the angles a^, a^... 
 tan a, — tan a„ 
 
 tan ajj — tan a. 
 
 .5 
 p ' 
 
 tana,-tana3 ^P ^^_ ^^_ 
 tan aj — tan a^ Pj 
 
72 THE FUNICULAR POLYGON. 
 
 If we simplify the problem by supposing the weights 
 Pj, Pj ... each equal to w, the equations (A) become 
 
 - = tan a, - tan a, = tan a - tan a^ = ... = tan a„ - tan a„^.i, 
 c ' 
 
 which results shew that the tangents of the angles a^,a^... 
 are in Arithmetical progression. 
 
 68. COE. If A CB be a heavy uniform string or chain sus- 
 pended from two points A, B; — C 
 the lowest point of the string, and 
 the angle which the tangent to it ■? j 
 at any point P makes with the \ // 
 horizon, we may obtain a simple \ /' 
 relation connecting ^ with the \. A" 
 length of the arc CP (= s). .....^V__,,^N 
 
 For we may regard the heavy ^ 
 
 string as made up of a series of 
 
 small equal weights attached at small equal intervals, and so 
 forming a funicular polygon: — and since the tangent at is 
 horizontal and the tangents of the angles which the succes- 
 sive elements of the string (taken from G) make with the 
 horizon are in Arithmetic progression, tan <^ will for different 
 positions of P vary as the number of the elements in the 
 arc CP, i. e. tan ^ oc s. 
 
 And further, if c, t be the tensions of the string at C 
 and P, we shall obtain for the conditions of equilibrium of 
 CP (which for this purpose we may regard as a rigid body) 
 
 s 
 t COS ^ = c, t sin (j> = s, and .'. tan (f> = - , 
 
 c 
 
 the weight of a unit of length of the string being here taken 
 as the unit of weight. 
 
( 73 ) 
 
 CHAPTER V. 
 
 OP THE CENTEE OF GRAVITY. 
 
 69. The attraction of the earth on any body would, 
 if unopposed, draw it towards the surface of the earth. 
 
 The direction in which a particle would fall freely at any 
 place is called the vertical line at that place. It coincides 
 with the direction of a plumb-line, or the normal to the 
 surface of standing water. 
 
 A plane perpendicular to this vertical line is said to be 
 horizontal. 
 
 If we regard the earth as a sphere (which is very nearly 
 the case), the vertical lines would all converge to the centre, 
 and therefore the directions of the forces which the earth 
 exerts on the different particles composing a body are not 
 parallel, strictly speaking. But since the dimension of any 
 body we shall have to consider is very small compared with 
 the radius of the earth, we may consider these directions to 
 be appreciably parallel, and the resultant attraction on the 
 body or system equal to the sum of the attractions on the 
 constituent particles; i.e. the weight of the whole equal to 
 the sum of the weights of the several parts. 
 
 The object of the present chapter is to shew that for 
 
 every body or system of particles there exists a point through 
 
 which the resultant attraction of the earth may be supposed 
 
 to act; i.e. a point at which we may suppose the weight of 
 
 \ the body to be collected, — a point whose position depends 
 
 ^only on the relative arrangement of the particles composing 
 
74 OF THE CENTRE OP GEAYITY. 
 
 the tody or system, and on the relative constitution of these 
 particles. If this point then were in rigid connexion with all 
 the parts of the system, all positions of the body or system 
 would be positions of equilibrium, if this point were supported. 
 
 Such a point in a body or system is called the centre of 
 gravity of the body or system, and we give the following 
 definition. — The point at which the weight of a body or 
 system may always be supposed to act, whatever be the 
 position of the body or system with respect to a horizontal 
 plane, is called the centre of gravity of the body. 
 
 70. We shall first shew that such a point exists in any 
 system of particles. 
 
 Prop. Every system of heavy particles has one and only 
 one centre of gravity. 
 
 First let us consider two heavy particles A, B, whose 
 weights are P, Q, and suppose them 
 connected by a rigid rod without 
 weight. Now, since P and Q act 
 
 through A and B in parallel direc- 4-^^ "^"^'^ 
 
 tions and towards the same parts, 
 they are equivalent to a single resultant, the magnitude of 
 which = P+ Q, and which acts through a point E in the line 
 AB, such that P : Q = BE : AE; and since the position of 
 E in the line AB does not at all involve the direction of 
 action of gravity, if this point E were supported, this system 
 of two particles would balance about E in any position. 
 E then is the centre of gravity of A, B, and the statical 
 effect of P and Q will be the same as if they were collected 
 into one particle and placed at E. 
 
DEFINITIONS, &C. 75 
 
 Again, if there are three particles A, B, C, whose weights 
 are P, Q, B, we can take E the centre of gravity of P, Q as 
 before, and suppose P+ Q placed at E instead of A and B, 
 and we then have two particles at E and G whose weights 
 are P+ Q and B; these then, as before, have a centre of 
 gravity at a point i^in the line EO, such that 
 
 P+Q : B=CF: FE, 
 
 and we may suppose P, Q, B all collected at F so far as 
 their statical effect is concerned. And so on whatever be the 
 number of particles, so that every system of heavy particles 
 has a centre of gravity. 
 
 Also a system of particles can have hut one centre of 
 gravity. For, if possible, let a system have two such points 
 G and G', and let the system be turned about if necessary 
 till the line joining G, G' is horizontal. Then we have the 
 weight of the system acting in a vertical line through G, and 
 also in another vertical line through G'; which is impossible, 
 since it cannot act in two different lines at the same time. 
 
 "We should arrive at the same point G in whatever order 
 we may take the points A, B, C... 
 
 COE. 1. Since every continuous body is an aggregation 
 of a great number of particles, every body has a centre of 
 gravity through which the resultant weight of the particles 
 acts: and we may suppose the weight of the whole body 
 collected at its centre of gravity. 
 
 And we may proceed to find the centre of gravity of a 
 system of bodies by supposing them to be a series of heavy 
 particles, the weights of which are equal to the weights of the 
 bodies, and which are in the position of the centres of gravity 
 of the several bodies. 
 
76 OF THK CENTRE OF GEAVITr 
 
 CoE. 2. The determination of the successive points E, 
 F, &c. in the previous proposition does not require the actual 
 weights P, Q, B, but only their ratios. Hence if the weights 
 of the several parts of a system be all diminished or all in- 
 creased in any the same proportion, the position of the centre 
 of gravity will not be altered. 
 
 CoK. 3. Since the weights P, Q, i?...are equivalent to 
 a series of parallel forces acting at the points A, B, C..., and 
 the position of the centre of gravity does not depend on the 
 direction in which these forces act, but only on their relative 
 magnitude and their points of application ; it would therefore 
 remain in the same position if the direction of these forces 
 were turned about their points of application in any manner, 
 still remaining parallel. Hence the point under consideration 
 is sometimes called the centre of parallel forces. 
 
 71. Having given the centre of gravity of a body and also 
 of a part of the hody, to find the centre of gravity of the re- 
 maining part. 
 
 Let w^, Wj be the weights of the two parts of the body; 
 G^, Q^ their respective centres of 
 gravity: — then G the centre of 
 gravity of the whole body must be 
 a point in the straight line which 
 joins G^ G^, such that 
 
 w,.GG, = w,.GG,. 
 
 Hence if G and G^ are given 
 in position, join G^ G and produce 
 
 it to (tj making GG^ = —^.GG^, and thus the position of G^ 
 
 the point required is determined. 
 
OF A EIGHT LINE. 77 
 
 72. Before proceeding to give a general method of finding 
 the centre of gravity of any system of particles, we will give 
 a few examples of finding the centre of gravity, — premising 
 that when we speak of a line, or plane, or surface as having 
 a centre of gravity, we suppose it to be made up of equal 
 particles of matter uniformly diffused over it: unless some 
 other supposition is stated. 
 
 I. To find the centre of gravity of a right line. 
 
 Considering it as a line of equal particles uniformly 
 arranged, it is clear that the middle point of the line is its 
 centre of gravity. For we may divide the line into a series 
 of pairs of equal elements, the particles composing any pair 
 being equidistant from the middle point. Hence the centre 
 of gravity of each pair is at the middle point, and therefore 
 the centre of gravity of the whole is there also. 
 
 ■ 
 
 II. To find the centre of gravity of a parallelogram. 
 
 Let ABCD be a parallelogram regarded as a uniform 
 lamina of matter, and draw the line , „ „ 
 
 EF parallel to AB or CD and bisect- ) T- 1 
 
 ing AD and 5(7,— and also the line F- /^ F 
 
 HK parallel to AD and bisecting I 1 / 
 
 AB and GD. The point G in which 
 
 HK, EF intersect is the centre of gravity required. For by 
 drawing lines parallel to BO and at equal distances from each 
 other, we may divide the parallelogram A G into a number 
 of equal small parallelograms whose lengths are all equal BG 
 and breadths as small as we please ; and we may take the 
 breadths so small that each may be regarded as a line of 
 particles, the centre of gravity of which is at its middle point. 
 
78 OP THE CENTEE OP GEAVITT 
 
 and which therefore is on the line EF, since EF bisects every 
 line that is parallel to BG. 
 
 Hence the centre of gravity of the whole parallelogram 
 lies in EF. Similarly it may be shewn to lie in HK, 
 
 Therefore G the point of intersection of EF, EK is the 
 centre of gravity of the parallelogram. 
 
 III. To find the centre of gravity of a plane triangle: 
 
 Let ABC be a plane triangular lamina of matter. From 
 any two of the angular points B, 0, 
 draw lines BF, CE bisecting the 
 opposite sides in F, E and cutting 
 each other in Q. Q is the centre of 
 gravity of the triangle. 
 
 By drawing a series of lines parallel -S" 
 to one of the sides AG at equal dis- 
 tances, we may divide the triangle into a number of quadri- 
 laterals, each of which, when their number is sufficiently 
 increased, may be regarded as a uniform material line. 
 
 Let ac be one such line cutting BF in /; then we have 
 af:AF=Bf:BF 
 = cf: GF; 
 by the two pairs of similar triangles afB, AFB and c/S, GFB. 
 Hence af:cf=AF: GF 
 = 1:1; 
 .-. af=^cf; i.e./is the middle point of ac, and is consequently 
 its centre of gravity. 
 
 Hence the centre of gravity of eacli of the lines composing 
 the triangle is in BF, and therefore the centre of gravity of 
 the triangle is in BF. 
 
OF A TEIANGLE. 79 
 
 Similarly the centre of gravity of the triangle may be 
 shewn to be in GE, whence we infer that G is the centre of 
 gravity required. 
 
 Further, if we join EF, 
 By similar triangles BQG, FGE; 
 BG: QF^BG-.EF 
 
 = BA : AEhj similar triangles AEF, ABG 
 = 2:1; 
 
 i. e. 5(? = 2 . GF; 
 .-. BF= 3 . GF; 
 i.e. GF=iBF, andBG = iBF. 
 
 In words, if a line be drawn from an angular point to the 
 middle of the opposite side, the centre of gravity of the 
 triangle lies on this line at a distance from the angular point 
 equal to two-thirds of the length of the line. 
 
 CoE. From this result it is easily seen that the centre of 
 gravity of the triangle coincides in position with that of three 
 equal particles placed at the angular points. 
 
 73. To find the centre of gravity of the "perimeter of a 
 _lriangle — regarding the sides as material lines of uniform 
 thickness. 
 
 Let A', E, G' be the middle 
 points of the sides of the proposed 
 triangle ABG — then the centre of 
 gravity of the perimeter ABG will 
 be in the same position as that of 
 three particles placed at A', B', G', 
 and whose weights are proportional. 
 to BG, G A, ABxQS^&civfAj. Draw 
 
80 OF THE CENTKE OF GEAVITY 
 
 A' a, B^ bisecting the angles A\ B' of the triangle A'BC, 
 then Euclid vi. 3, 
 
 B'a : C'a = A'B' : ^' C" = AB : A 0. 
 
 Hence a is the centre of gravity of the two sides AB, 
 AC, and therefore the centre of gravity of the whole peri- 
 meter lies in the line A' a, — similarly it lies in the line B'^, 
 — the centre of gravity required must therefore be the point of 
 intersection of these two lines — which is the centre of the 
 circle inscribed in the triangle A'B' C. 
 
 74. Having shewn that every system of particles has 
 one and only one centre of gravity, we proceed to shew how 
 to find it in any case ; 
 
 (i) for a series of particles lying in a straight line. 
 
 (ii) in one plane. 
 
 (iii) arranged in any manner in space. 
 
 I. To find the, centre of gravity of a series of heavy 
 particles lying in a straight line. 
 
 Let A, B, C... be the several 
 
 particles whose weights are JP, Q, \ 4 P c p 
 
 B. . . and lying in the straight line ^' ^^ 
 
 Ox. Let be a fixed point in the line, and let x^, x^,x^... 
 be the distances of the particles A, B, C..from 0; then if^ 
 be the centre of gravity of A and B, 
 
 P:Q^Bg^:Ag^, 
 
 or P.Ag^=Q.Bg,; i.e. P{Og^-x^)= Q{x^- Og^, 
 
 i.e. {P+Q) Og, = Px^+Qx, (i) ; 
 
 a result which we might have obtained at once from the 
 
OP A SYSTEM OF POINTS. 
 
 81 
 
 consideration that since tte resultant of the forces ^and Q at 
 A and 5 passes through g^, the sum of the moments of 
 P and Q about any point is equal to the moment of their 
 resultant P+ Q. 
 
 Again, considering P and Q as collected at g^, iig^ be the 
 centre of gravity of P+ ^ at ^'^ and B at C, we have as 
 before 
 
 (P+ Q + B) Og,= {P+ Q) Og^ + Bx, 
 
 =^Px,+ Qx, + Bx^, by (i) (ii). 
 
 Similarly {P+ Q+B+8)0g,= iP+ Q+B) 0g,+8x, 
 
 = Px^+ Qx^+Bxg+ 8x^... (iii) . 
 
 And so on for any number of particles. 
 
 Hence if we call x the distance of G the centre of gravity 
 of the whole from 0, 
 
 - Px,+ Qx^ + Bx^+Sx^+... _l<{Px) 
 X- „. ^ . „ . 2^p^ 
 
 .(iv). 
 
 P+Q + B+... 
 
 The centre of gravity then is in the same line as the 
 particles, and the distance of it from any assumed point is 
 given by (iv). 
 
 II, To find the, centre, of gravity of a series of heavy 
 particles lying in one plane. 
 
 Let A, B, C..be the system of 
 particles whose weights are P, Q, 
 B. . . and let them be referred to two 
 axes Ox, Oy at right angles to one 
 another in the plane in which the 
 particles are. Join AB, and take 
 ^j the centre of gravity of P and ^ 
 P. M. 
 
 M "' -A; ^iVM 
 
 6 
 
82 
 
 OP THE CENTRE OF GRAVITY 
 
 at A and B so that P: Q = Bg^'.Ag^. Join g,0, and take 
 g^ the centre of gravity of P+ ^ at ^-^ and 2? at C so that 
 
 P+Q'.R=Cg^:g^gi, 
 
 and so on till we find G the centre of gravity of the whole, as 
 in Art. 70. Draw AN^, g^n^, BN^...^aiXsX\&\ to Oy, meeting 
 Ox in N^, «!, ... 
 
 If now we call 
 
 AN=yy BN, = y, 
 
 &c. 
 
 OV=x\ 
 
 av=y] 
 
 our object is to find x, y which determine the position of Q, 
 in terms of x^y^... and P, Q... 
 
 Now, considering g^ the centre of gravity of A and B, we 
 have 
 
 P.Ag=Q.Bg, (i) ; 
 
 and if through A and g^ we draw two j^ 
 lines parallel to NJSf^ we should have 
 two similar triangles; comparing the 
 sides of which we get 
 
 ^9. •• -%t = 5'i<-^-^i : BN,-g^n^ (ii), 
 
 whence from (i) and (ii) 
 
 P. {3,n,-AN,) = Q.{BN,-g,n,), 
 or {P+Q)g^n^ = P.AN,+ Q.BN, = P.y^+Q.y^...{m); 
 now introducing a third particle G we have similarly 
 {P+Q + B) g,n, = (P+ Q) g,n, + B . GN,, 
 
 = P.y,+ Q.y, + B.y, (iv), 
 
 and so on whatever be the number of particles y 
 
 I.e. y 
 
 ^ ^^,_ P.y,+ Q.y,+ ... _ tiPy) 
 
 P+Q + 
 
 S(P) 
 
 w. 
 
OP A SYSTEM OF POINTS. 
 
 By a similar mode of proceeding we shall obtain 
 
 ^=^(^-^ 
 
 t{,Jfi 
 
 .(Yi). 
 
 These two results (v) and (vi) determine the position of the 
 centre of gravity of the system of particles, which lies in the 
 plane of the particles. 
 
 III. To find the centre of gravity of a system of particles 
 arranged in any manner in space, 
 
 ' Let the system of particles 
 A, B, G... whose weights are P, 
 Q, R... be referred to three lines 
 Ox, Oy, Oz mutually at right 
 angles ; 
 
 let g^ be the C.G. of A and B. 
 
 g^ -4,5, andC,&c. 
 
 Through A B G ... g^g^ . . . draw 
 
 AJSr^, BN^...gji^, g^n^ parallel 
 
 to Oz meeting the plane xOy in' 
 
 N'^,N'^...n^,n^...a,nd. through these 
 
 points draw in the plane xOy the lines N^^M^, NJH^.-.n^m^, 
 
 n^^... parallel to Oy meeting Ox in M^, M^... 
 
 If now OM^ = xA 
 
 M^N^ = yS and similar quantities for each particle, 
 N^A = z,\ 
 
 and ii xy z be the corresponding quantities for G, the centre 
 of gravity of the system, — ^we have, considering A and B only 
 at first, 
 
 P.Ag^=Q.Bg^; 
 
 6—2 
 
84 OF THK CENTRE OF GKAVITT. 
 
 or if we draw lines through A^, g^ parallel to NJil^, we have 
 by similar triangles 
 
 Ag,i Bg, = g^n^ - AN, : BN, - g,n,, 
 
 whence P . {g,n, - AN^) = Q . (BN^ - g^n^} ; 
 i.e. {P+Q).g,n,= P.AN,+ Q.BN, 
 
 similarly introducing another particle 0, g^ being the centre 
 of gravity of A, B, 0, and therefore the centre of gravity of 
 P+ Q at ^j and B at C; 
 
 {P+ Q + R)g,n= (P+ Q)g,n,+R. CN, 
 = P.z,+ Q.z, + B.z„ 
 and so on for any number of particles — till we get 
 {P+Q + ...) aV=P.z, + Q.z^+... 
 - P.z^+Q.z,+ ... _ t{Pz) 
 
 °^^- P+Q+... S(Pr 
 
 we should similarly have 
 
 - S(Py) ,- 2 (Pa;) 
 
 2'=2(:^^"^''=-(2py 
 
 These three expressions for xi/ z determine the position 
 of the centre of gravity of the system of particles considered. 
 This includes I. and II. as particular cases. 
 
 75. Ohs. In the case III. of the preceding article it will 
 in general be convenient to take the lines Ox, Oy, Oz at right 
 angles, but the student will observe that the course of the 
 proof does not require that the lines Ox, Oy, Oz should be 
 inclined at any particular angles : he may then in any par- 
 
GENERAL REMAEKS. 85 
 
 ticular case assume three lines (not in one plane) inclined at 
 any angles which may appear to him most convenient in the 
 case under his consideration; — and a similar remark applies to 
 case II. 
 
 Def. The moment of a force with respect to a plane is the 
 product of th.Q force into the distance of its point of application 
 from the plane. If the points of application of two fprces 
 are on opposite sides of a given plane, the moments of the 
 forces with respect to that plane will have opposite signs. 
 This must he carefully distinguished from the moment of a 
 force with respect to a, point or an axis. Art. 31. 
 
 COE. 1. We see from the results of Art. 74, that the alge- 
 braic sum of the moments of the particles of a system with 
 respect to any plane is equal to the moment of the whole 
 (supposed to be collected at the centre of gravity) with re- 
 spect to the same plane. 
 
 From whence follows the conclusion, that if the algebraic 
 sum of the moments of a system taken with respect to any 
 proposed plane be zero, the centre of gravity of the system 
 lies in that plane; and vice versa, if the centre of gravity of a 
 system lie in a given plane, the algebraic sum of the mo- 
 ments of the particles with respect to that plane is zero, — or 
 in other words, the sum of the moments of the particles which 
 are on one side of the plane is equal to the sum of the mo- 
 ments of the particles which are on the other side of the 
 plane. 
 
 CoE. 2. If we suppose a system to be divided into any 
 number n of particles of equal weights we have the distance 
 
 of centre of gravity from any plane = -th the sum of the 
 
86 OF THE CENTRE OP GRAVITY. 
 
 distances of all the particles from the same plane. Viewed 
 in this manner, the centre of gravity of a body or system is 
 sometimes called the centre of mean position of the body or 
 system, or the centre of figure. 
 
 Cor. 3. If a system of particles be projected on any 
 plane, the projection of the centre of gravity of the system 
 on that plane will be the centre of gravity of a system of 
 particles in the plane, equal to the former and coincident 
 with the points of projection of the original system. 
 
 This appears at once from the results of Art. (74), for the 
 values oi xy z depend only on the weights of the particles 
 and their distances estimated parallel to Ox, Oy, Oz from the 
 planes yOz, zOx, xOy severally. 
 
 76. Centre of parallel forces. 
 
 If in any of the cases of Art. (74), A, B, G... be the point 
 of application of a system of parallel forces P, Q, B,... the 
 method pursued in that article will lead to formula for the 
 co-ordinates of the point of application of the resultant of such 
 a system of parallel forces, viz. 
 
 ^_ ^iPx) ' ^ %{Py) - t{Pz) 
 '"~S(P)' ^~2(P)' ^~t{P) ^'^' 
 
 in the most general case. 
 
 These results are algebraically true whether the forces act 
 all in the same direction or not — and we may interpret them 
 as stating that the resultant of a system of parallel forces 
 is = S (P) acting at a point whose co-ordinates are given by 
 equation (i). 
 
GENERAL EEMAEKS. 87 
 
 If however "t (P) = 0, and the expressions S [Px), % {Py), 
 A (Pa) do not each = also, the system will be equivalent 
 to a couple which does not admit of heing represented by a 
 single resultant force, Art. (30). 
 
 77. The position of the centre of gravity of a body or 
 a system of particles depends (as we have seen, Art. 74) only 
 on two things; (i), the form of the body, or, in other words, 
 the arrangement of the particles of the system ; and (ii), the 
 relative density of the different parts. 
 
 Formulae have been obtained in Art. 74, by which the 
 centre of gravity of any system of particles whose relative 
 weights and position are known, may be found ; and we have 
 seen in Cor. 1, Art. 70, that a body may be considered as 
 a particle placed at the centre of gravity of the body, so 
 that if the fcentres of gravity of the several bodies composing 
 a system be known, we are enabled to find the centre of 
 gravity of the system, and the problem assumes a general 
 character. 
 
 The determination however of the centre of gravity of a 
 body (either a continuous solid body, or a surface regarded 
 as a lamina of matter of indefinitely small thickness) will in 
 general require the aid of the Integral Calculus. 
 
 Obs. Cases will not unfrequently arise in which the 
 position of the centre of gravity can be assigned from geo- 
 metrical considerations such as the following, which are 
 suggested for the consideration of the student. 
 
 1°. If m any body or system a plane can be found which 
 divides the body into two parts which are symmetrical with 
 respect to the plane on opposite sides of it, the centre of 
 gravity of the body must lie in that plane. 
 
88 OF THE CENTRE OF GRAVITY 
 
 For since the Tbody is divided symmetrically into two 
 parts, these parts must be equal, and their centres of gravity 
 at equal distances from the plane on opposite sides of it. 
 Hence the centre of gravity of the whole, which is the middle 
 point of the line joining the centre of gravity of the two 
 parts, must lie in the plane under consideration. 
 
 2°. Hence it follows readily, that if three planes can he 
 assigned, each of which divides the body or system symme- 
 trically into two parts, the common point of intersection of 
 the planes is the centre of gravity of the body. 
 
 3°. Observation 1° applies to all bodies or systems of 
 bodies of uniform density; it is also true if the densities are 
 not uniform, provided the densities of all elements of the 
 body symmetrically situated on opposite sides ©f the plane 
 are severally the same. The same may be said of curved 
 surfaces. But in the case of plane areas we need only con- 
 sider lines in its plane which divide the area symmetrically, 
 and we may assert (with a proof similar to that of 1°), that 
 in any plane area if a line can be found which divides it 
 into symmetrical parts, the centre of gravity lies in that 
 line ; and further, if two such lines can be found their poiat 
 of intersection is the centre of gravity of the area. 
 
 The same remarks apply in this case as in that of a body, 
 if the density of the area be not uniform. 
 
 78. Some conclusions arising from these observations, 
 1°, 2°, 3°, are the following. 
 
 (i) The centre of gravity of a right line is its middle 
 point. 
 
OP A PTEAMID. 89 
 
 (ii) The centre of gravity of a parallelogram is the 
 intersection of its two diagonals ; in other words, the middle 
 point of one of them. 
 
 (iii) The centre of gravity of a solid parallelepiped, or 
 of the surface of a, parallelopiped, is the intersection of its 
 four diagonals, which is the middle point of any one of them. 
 
 (iv) The centre of gravity of a circular area, or of a 
 circular ring, is the centre of the circle. 
 
 And that of a solid sphere, or a spherical surface, or sphe- 
 rical shell, is the centre of the sphere. 
 
 These results will be of frequent use. 
 
 79. To find the, centre of gravity of a pyramid on a 
 triangular base. 
 
 Let ABC he the base of the pyramid, and Fits vertex. 
 
 Take D the middle point of 
 one of the sides BG, and join AB, 
 VB, in which take E and S such 
 that AE = iAB and FJ3"=|FP, 
 (and HE is therefore parallel to 
 A V) ; then E, H are the centres of 
 gravity of the triangles ABG, VBG; 
 if now we join VE, AH, they will 
 intersect in some point G, since 
 they both lie in the plane A VB. 
 
 Q is the centre of gravity of the pyramid. 
 
 For suppose the pyramid to be made up of an indefinite 
 
90 OF THE CENTEE OF GKAVITY. 
 
 number of thin triangular plates all parallel and similar to 
 ABO, and let abc be any one of these; 
 
 if VB meet be in d, and VS meet ad in g, we have by 
 similar triangles, 
 
 ag : AE= Vg : VE=gd : ED; 
 
 .'. ag : gd=AE : EJ) = 2 : 1. 
 
 Hence since d is the middle point of be, g is the centre of 
 gravity of the plate abc. 
 
 Similarly it may be shewn that the centres of gravity of 
 all the plates of which the pyramid is composed lie in the 
 line VE. 
 
 And in a similar way by supposing the pyramid made up 
 of plates parallel to VBC, the centre of gravity of the whole 
 may be shewn to lie in AH. 
 
 Hence G the point of intersection of VE, AH is the centre 
 of gravity of the pyramid. 
 
 Further if we join SE which will be parallel to A V, we 
 have by similar triangles A VG, HGE. 
 
 VG AV AD . „ 
 
 GE=TE=ED'^^' " VG-3.GE, 
 
 .: VE=iGE, or EG = iVE, and .-. VG=-IVE; 
 
 i. e. if the vertex be joined with the centre of gravity of the 
 base, the centre of gravity of the pyramid is a point in this 
 line at a distance of Jths of it from the vertex, and Jth of it 
 from the base. 
 
GENERAL THEOREMS. 91 
 
 Cor. 1. To find the centre of gravity of any pyramid 
 whose hose is a plane polygon. 
 
 Join V the vertex with the centre of gravity of the 
 base, and in this line take a point G at 
 a distance from the base equal to Jth of 
 the length of the line. G shall be the 
 centre of gravity of the pyramid. For 
 it may be shewn as in the present article, 
 by supposing the pyramid to be made 
 up of plates parallel to the base, that 
 the centre of gravity of the pyramid lies 
 in this line. 
 
 And again, by dividing the base into triangles the py- 
 ramid may be divided into a series of triangular pyramids 
 having a common vertex: and if we draw a plane through G 
 parallel to the base, this plane will contain the centres of 
 gravity of all the triangular pyramids, since it would cut the 
 line which joins the vertex with the centre of gravity of the . 
 base of any of the triangular pyramids in a point whose 
 distance from the base is Jth of the length of the line. 
 
 Since then the centres of gravity of aU the triangular 
 pyramids lie in this plane, and it has been shewn to lie 
 in the line VO, G must be the centre of gravity of the 
 pyramid. 
 
 Cor. 2. Since a curve may be regarded as the limit of a 
 polygon, whose sides are indefinitely increased in number 
 and diminished in magnitude, we may consider a cone on any 
 base as the limit of a pyramid, and its centre of gravity will 
 be in the line joining the vertex with the centre of gravity of 
 
92 OF THE CENTEE OF GEAVITT. 
 
 the base, at a distance from -the vertex equal to f ths of this 
 line. 
 
 If the cone be a right cone on a circular base, the centre 
 of gravity is in the axis of the cone, at a distance from the 
 vertex equal to f ths of its length. 
 
 Cor. 3. The centre of gravity of a triangular pyramid 
 coincides in position with the centre of gravity of four equal 
 heavy particles placed at its angular points. 
 
 For we easily see by the construction that E is the centre 
 of gravity of three equal particles P placed at A, B, 0, and 
 G will be the centre of gravity of 3P at H, and P at V, since 
 GV: EV=3 : 4. 
 
 COE. 4. We can proceed to find the centre of gravity of 
 any solid bounded by plane faces. For we may divide the 
 solid into a series of pyramids, the centre of gravity of each 
 of which can be found, and if we suppose at each of these 
 points weights to be placed proportional to the several pyra- 
 mids, the centre of gravity of these weights will coincide with 
 the centre of gravity of the solid. 
 
 Similarly with any plane area bounded by straight lines, 
 by dividing it into a series of triangles, and supposing par- 
 ticles placed at the centre of gravity of each triangle pro- 
 portional to the areas of the triangles, the centre of gravity 
 of these particles will be the centre of gravity of the area. 
 
 80. Before concluding this chapter we will give a few 
 general theorems relating to the centre of gravity. 
 
 I. If a hody he suspended from a point about which it can 
 swing freely, it will rest with its centre of gravity in the verti- 
 cal Une which passes through the point of suspension. 
 
STABLE AND UNSTABLE EQUILIBRIUM. 
 
 93 
 
 Let AC he tlie body, G its centre of gravity; and S the 
 point of suspension. Draw G V ver- 
 tical, and /SF horizontal to meet GV (" 
 in F; then the only forces which act 
 on the body are its weight, which acts 
 in the vertical line VG, and the reac- 
 tion arising from the fixed point B. 
 
 These two forces cannot balance 
 each other (and consequently the 
 body cannot be at rest) unless they 
 act in the same line in opposite directions, i.e. unless VG 
 pass through B. 
 
 i.e. the body cannot be at rest unless the vertical line 
 through G pass through B; and when this is the case, the 
 fixed point will exert a force on the body sufficient to balance 
 the weight of the body and therefore equal and opposite to 
 that weight. 
 
 Or we migTit reason thus. When a body is at rest 
 under the action of forces in one plane, the moments of the 
 forces about any point vanish : but in this case, if we take 
 the moments about B, the weight of the body has a moment 
 about 8 = weight y. BV, which is not counterbalanced by any 
 other moment, and this cannot vanish imless 8V= 0, i. e. 
 unless the line joining B and G is vertical. "Whence the 
 same conclusion as before. 
 
 CoE. This proposition leads to a mode of determining 
 the centre of gravity of a body which may sometimes be 
 practically available, thus, — Let the body be suspended freely 
 from any points of its surface in succession, and let the line 
 in the body which is vertical and passes through the point 
 
94 
 
 OF THE CENTEE OF GEAVITT. 
 
 of suspension be noted in each case, — the point of intersec- 
 tion of two such lines is the centre of gravity sought, 
 
 81. In the proposition, G may either be directly above 
 or below 8 when in a position 
 of equilibrium; but the nature of 
 the equilibrium is very different in 
 the two cases. In fig. 2 if the body 
 be slightly displaced by turning it 
 about 8 through a small angle, it 
 is evident G would be raised, and 
 if the body be then left to the action 
 of gravity, its first tendency would 
 be to return towards its former po- 
 sition of equilibrium. 
 
 But in figure 1 if the body were slightly displaced by 
 being turned about 8 through a small angle, the tendency of 
 the body would be to recede further and further from its 
 position of equilibrium. 
 
 The above are simple cases of equilibrium, which are 
 called stable and unstable respectively; the meaning of which 
 the student will understand from the following definition. 
 
 Def, When a body is in equilibrium under the action of 
 a system of forces, if the body be slightly displaced the action 
 of the forces on the body in its new position will in general tend 
 either to make it return towards or recede from its original 
 position of equilibrium; in the former case the equilibrium is 
 said to be stable, or the body to be in a position' of stable equi- 
 librium; in the latter, the equilibrium is said to be unstable, 
 or the body is said to be in a position of unstable equilibrium. 
 
STABLE AND UNSTABLE EQUILIBKIDM. 
 
 95 
 
 We say in general, because the above is not always the 
 case, for in certain cases the forces in the new position of the 
 body may still have no tendency to make the body move one 
 way or the other; a position of this kind is called one of 
 neutral equilibrium — as in the case of a sphere resting on a 
 horizontal table. 
 
 Or again, the forces in the new position may tend to make 
 the body neither return to its former position nor recede from 
 it, but to give it a rocking or rolling motion; as in the case of 
 an ellipsoid resting on a horizontal plane at the extremity of 
 its mean axis. 
 
 82. II. A hody placed on a horizontal plane will 
 stand or fall over, according as the vertical line dravm 
 through the centre of gravity of the hody falls within or 
 without the base. 
 
 Let A GBD be the base of the body in contact with the 
 plane, GE the vertical line drawn 
 through the centre of gravity of the 
 body and meeting the base in some 
 point E within it. 
 
 Now the pressure which the 
 weight of the body exercises on the 
 plane is equal to a weight Tf acting 
 in OE. 
 
 And if E lies within the base, 
 the plane will be capable of exercising a vertical pressure 
 passing through E of sufficient magnitude just to balance W; 
 and the body will be in equilibrium. 
 
96 
 
 OF THE CENTRE OP GRAVITY. 
 
 But if E fair with out the base the plane cannot exert a 
 pressure which shall pass through 
 E and balance W: in this case 
 then the body will not be in equi- 
 librium, but will begin to fall over 
 by turning round some tangent line 
 to the perimeter of the base, and this 
 will obviously be about the point of 
 the base which is nearest to E. ' 
 
 Ohs, By the base here is meant 
 the extreme polygon formed by 
 
 joining all the points of contact of the base^or the area 
 enclosed by a string drawn tightly about the base. 
 
 Cor. 1. In a similar manner it may be shewn that if a 
 body be placed on an inclined plane and it be prevented from 
 sliding along the plane by friction or otherwise, the body 
 will stand or fall over according as a vertical line drawn 
 through the centre of gravity of the body falls within or 
 without the base. 
 
 Cor. 2. In figure (1) if an effort were made to make the 
 body turn about some point A in the perimeter of its base, 
 the moment about A of the force employed must be at least 
 equal to the moment of the weight of the body about A; 
 which moment is = W.AE. This moment then measures 
 the effort necessary to make the body fall over; and it is 
 clear that the less AE is, the less effort will be required. If 
 AE=0, the moment vanishes, and any the slightest effort 
 would make the body fall over. This accounts for the diffi- 
 culty of making a body balance about a point immediately 
 under the centre of gravity. 
 
STABLE AND UNSTABLE EQUILIBEIUM. 
 
 97 
 
 Compare witli this the remarks on stable and unstable 
 equilibrium, in the previous article. 
 
 c;....~- 
 
 83. When a rough body BA G rests upon another PA Q 
 fixed — the surfaces near A the 
 point of contact being spherical — 
 the condition of the stability or 
 instability of the equilibrium may- 
 be simply investigated thus. 
 
 The common normal to the 
 two surfaces at A will be vertical 
 and will pass through 0, 0^ the 
 centres of the spherical surfaces 
 of BA C, QAR, £md also through 
 Q the centre of gravity of BA G. 
 Let BAG be displaced by rolling 
 through a small angle so as to 
 
 come into the position BA'C — through P. the new point of 
 contact draw Pif vertical, meeting A! 0' in M. Then accord- 
 ing as A!G' is < or > A!M, the weight of B'A'G' will tend 
 to make it return towards or recede further from its original 
 position of equilibrium by turning about the point of contact 
 p — that is, the equilibrium will be stable or unstable respec- 
 tively. 
 
 Let AO = r, A0^ = R, AG = h, ^ AO,P = = MPO', 
 A' 0'P= (p, so that r^ = Ed since the arc AP= arc A P. 
 
 Now 
 
 O'M O'M 
 
 sin d 
 
 sin 6 
 
 OP ,m{9 + 4>) ^.^fR+zA li 
 
 + r 
 
 in the limit when is taken very small ; 
 
 P.M. 
 
98 OF THE CENTRE OF GEAVITT 
 
 .-. 0'M=- 
 
 .: A'M=r-0'M=r- 
 
 JX + r' 
 
 Rr 
 
 M + r E+r' 
 and the equilibrium is stable or unstable according as h is 
 
 < or > A'M, i. e. ho -73 . 
 
 it + r 
 
 Or, as it may be written, 
 
 h r U 
 Ohs. If G', M coincide — the displacement being very 
 small, — in which case x = -+ p — the equilibrium is said to 
 be neutral. 
 
 h' 
 
 "B 
 
 84. Ols. (i) If the sur- 
 face QAB, be concave, we may- 
 change the sign of R, and we 
 shall have the equilibrium 
 stoihle or unstable according 1" 
 as 
 
 1 1 1 
 
 (ii) If the surface of BA be plane — as in the case of 
 a solid resting with its plane base 
 upon a curved surface — r = co , and the 
 equilibrium is stable or unstable ac- 
 cording as 
 
 A < or > ^. 
 
 (iii) If the surface of QAR be 
 plane — as in the case of a solid resting 
 
OF A CIECULAE AEC. 
 
 99 
 
 •with its curved surface upon a horizontal plane — 5 = 00, 
 and the equilibrium will be staile 
 or unstable according as 
 
 h <ox> r. 
 The above particular cases 
 (i), (ii), (iii) of the general one, (^ ^ f" [ ^ 
 
 may be investigated independ- ' ^ 
 
 ently by the student. 
 
 85. The following is an example of finding the centre 
 of gravity which leads to some useful results. 
 
 To find the centre of gravity of n equal particles arranged 
 at equal intervals along a circular arc. 
 
 Let be the centre of the circular are AB, along which 
 the n equal particles A, P, Q, 
 B, ... £ axe arranged; 
 
 ^A0B=2a, AO = a, 
 
 e = ^AOP=^FOQ=... 
 
 so that {n-l)d = 2a (i). 
 
 Then if {x^,y^ {x^,y^... 
 be the co-ordinates of the suc- 
 cessive particles A, P, Q... re- 
 ferred to Ox, Oy as rectangular axes, we have (Art. 74) 
 
 y 
 
 I 
 
 > 
 
 
 
 
 
 
 
 A 
 
 JB 
 
 - t{Px 
 
 ■.-I. 
 
 {x^ + x^+...+x„} 
 
 S(P) 
 
 - (1 + cos ^ + cos 2^ + . . . + COS (n - 1) 6] 
 
 n — l„ . n „ 
 asm- 9 
 
 (by Trigonometry) 
 
 cos- 
 
 sin|^ 
 
 7—2 
 
100 OP THE CENTRE OF GEATITY 
 
 n 
 
 COS asm a. 
 
 a n — l 
 
 n . a. 
 
 sm- 
 
 w-1 
 
 by substituting for 6 in terms of a from (i). 
 
 , , - t (Pv) 1 , , 
 
 And y = -^^ =-Wi + i/A "■'+ y»l 
 
 = - {sin d + sin 20 + ... + sin (n - 1) 0] 
 
 . n — l„ . n „ 
 
 sm 6 &va.-a 
 
 a 2 2 
 
 n 
 
 sin ^6 
 
 n 
 
 sm a sm a 
 
 a n — l 
 
 n . OL 
 sm- 
 
 .(ii) 
 
 If G be tbe centre of grayitj, 
 
 n 
 sm- a 
 
 OG = ^/W+f=- """^ (iii), 
 
 sm zr 
 
 n — 1 
 
 and tan AOG = ~^ = tan a, 
 
 a; 
 
 i. e. G lies in the line OG which bisects the i A OB, and (iii) 
 gives its distance from 0. 
 
 86. Cob. From the preceding investigation we may- 
 deduce some useful result?. 
 
OF A CIECULAE AKC. 
 
 101 
 
 If the number of particles n be supposed to become in- 
 definitely great, 
 no. 
 
 , becomes = a, and n sin - 
 n — 1 ) 
 
 becomes = a 
 
 in the limit, 
 and in this case 
 
 oa=a 
 
 sma 
 
 I. Since a uniform material circular arc may be regarded 
 as a series of equal particles at small 
 equal intervals, — ^if AB be a uniform 
 circular arc of which is the centre, 
 and G the centre of gravity, 2a the 
 circular measure of the ^AOB and 
 AO = a\ 
 
 then we infer from the above that 
 OG* bisects the/ ^05, 
 
 sin a 
 
 and 
 
 OG = a- 
 
 II. Again, since we may regard the circular arc A£ as 
 
 the limit of a polygon of a very large 
 
 number of sides, we may regard the 
 
 circular sector A OB as made up of a 
 
 very large number of triangles having 
 
 a common vertex at 0, and the sides 
 
 of this polygon for their bases, — and 
 
 if Or be the distance from of the 
 
 centre of gravity of any one of these 
 
 triangles Ojpq, we shall have (when 
 
 the ipOqis taken very small) 
 
 2 2 . 
 
 Or = -Op=^a, in the limit, 
 
 o o ■ 
 
102 
 
 OF THE CENTEE OP GEAVITT. 
 
 and the centre of gravity g of the sectm- A OB will coincide 
 with the centre of gravity of a uniform circular arc cih whose 
 
 radius =5.0. 
 a 
 
 i. e. Og bisects the / A OB, and Og= -.a . 
 
 If a = — the sector becomes a semicircle, and In this case 
 
 III. The centre of gravity G of 
 the sector A OB being known, as well 
 as (?j that of the triangle A OB, — we 
 can easily (Art. 71) find G^ the cen- 
 tre of gravity of the circular segment 
 ABC. 
 
 For A A OB = a^ sin a cos a, 
 
 sector A OB = a\ 
 segment ABC= a" (a — sin a cos a). 
 
 AT nn 2 ^^ 2 sina 
 
 Also C'G',=-o cos a, OG = -a ; 
 
 '3 3 a 
 
 „ , . \ r\^ , 2 sin a 
 
 .•. a (a — sma coso). C'G^„=aa.-a 
 
 ^ ' ^ 3 a 
 
 • a sm a cos a..— a cos a. 
 
 = - a" (sin a — sin a cos^ a) 
 
 = - a sm a ; 
 o 
 
 •• 0G=-. 
 
 sin* a 
 
 ' 3 a — sin a cos a ' 
 which determines the c. G. of the segment. 
 
LEIBNITZ THEOREM. 
 
 103 
 
 IV. The centre of gravity Q 
 of a solid hemisphere ABC lies ia 
 the radius 0(7 which is perpendicu- 
 
 lar to the base, and 0Q = -.00. 
 
 8 
 
 Also, the centre of gravity of the 
 hemispherical surface ABO bisects 
 OC. 
 
 These results may be obtained 
 by processes similar to those employed in this article — but 
 much more easily by employing the Integral Calculus : — we 
 have therefore thought it sufficient to state the results for the 
 information of the student. 
 
 87. We will close this chapter with the following 
 elegant theorem — due we believe to Leibnitz. 
 
 III. ^a system of forces in equilibrium acting at a point 
 A be represented in magnitud,e and direction by the lines AP, 
 AQ, AR... then will the point A be the centre of mean position 
 of the points P, Q, K, ... ; {in other words) the point A will 
 be the centre of gravity of a system, of equal particles placed 
 at the points P, Q, H... 
 
 Take any line siiAx passing through A and draw Pp, 
 Qq ... perpendicular to this line ; 
 then -sr'^ Ap, Aq... represent the 
 projections on aiAx of the lines 
 AP, AQ...'\.&. of the forces P, Q... 
 But since these forces are at equi- 
 librium the algebraic sum of their 
 resolved parts in any assigned 
 direction must be zero by Art. (39), 
 Hence since the algebraic sum of the lines Ap, Aq ... is zero, 
 
104 LEIBNITZ' THEOKEM. 
 
 the centre of gravity of the points P, Q... must be in the plane 
 ■which passes through A at right angles to x'Ax, and since 
 the direction of x'Ax is arbitrary, this centre of gravity must 
 lie in every plane which can be so drawn, and must therefore 
 coincide with the point A, the common point of intersection 
 of these planes. 
 
 Hence, when any number of forces acting on a point are 
 in equilibrium, this point is the centre of gravity of a series 
 of equal particles placed at the extremities of lines which 
 represent the forces in magnitude and direction. 
 
 And vice versd. If we consider a series of equal particles 
 and we draw lines from each ,to the centre of gravity of the 
 series, it is clear that a system of forces represented by these 
 lines will be in equilibrium. 
 
 For as before draw the lines AP, AQ...; it ia clear that A 
 being the centre of gravity, the algebraic sum of the lines Ap, 
 Aq. . . is zero ; i. e. the sum of the resolved parts of the forces 
 AP, AQ... taken in any direction xAx is zero, and therefore 
 the forces are in equilibrium. 
 
 Cor. 1. We see from this theorem that if three forces are 
 in equilibrium about a point, this point is the centre of gravity 
 of the triangle formed by joining the extremities of lines 
 representing the forces in magnitude and direction; for the 
 centre of gravity of a triangle is the same as that of three equal 
 particles placed at its angular points. 
 
 Similarly, if four forces are in equilibrium about a point, 
 this point is the centre of gravity of the pyramid whose 
 angular points are the extremities of the straight lines 
 
LEIBNITZ' THEOEEM. 105 
 
 representing the forces : for the centre of gravity of a trian- 
 gular pyramid is in the same position as that of four equal 
 particles placed at the angular points. 
 
 The converse of each of these is also true. 
 
 COE. 2. More generally: If all the equal particles of a 
 rigid body of any form are attracted to the same point by 
 forces proportional to their distances from this point they will 
 be in equilibrium if the point be the centre of gravity of the 
 body ; — and conversely. 
 
( 106 ) 
 CHAPTEE VI. 
 
 OF THE MECHANICAL POWERS. 
 
 88. The simplest machines employed for supporting 
 weights, communicating motion to bodies, — or speaking gene- 
 rally, for making a force which is applied at one point prac- 
 tically availahle at some other point, are called the Mechanical 
 Powers ; and by a combination of them all machines, however 
 complicated, are constructed. 
 
 They are commonly reckoned as six in number: — the 
 lever, the wheel and axle, the jaully, the inclined plane, the 
 wedge, and the screw. 
 
 In explaining and discussing these simple machines we 
 shall suppose them to be at rest, so that the force applied at 
 one point is balanced by the force or pressure called into 
 action at some other point : we shall also suppose the several 
 parts of them to be without weight and perfectly smooth 
 except when the contrary is expressly stated. 
 
 When two forces acting on a machine balance each other, 
 one of them is for convenience called the power and the other 
 the weight. 
 
 89. The Lever. 
 
 A rigid rod or bar capable of turning about a fixed 
 point of it is called a lever. The point about which it can 
 turn is called the fulcrum, and the parts into which the rod 
 is divided by the fulcrum are called the arms of the lever. 
 When the arms are in a straight line, it is called a straight 
 lever; in all other cases it is a lent lever. 
 
THE LKVEE. 107 
 
 We have seen in Art. 35, 36, that a body of any form 
 capable of turning about a fixed 
 point may be considered as a 
 lever, and if two forces P, Q act 
 upon it in a plane passing through 
 0, the lever will be in equilibrium 
 if P. Op=:Q . Oq; i.e. if the 
 moments of P and Q which tend 
 to turn the lever about be equal, 
 and tend in apposite directions. 
 
 In order however to render our explanation as simple as 
 possible, we will for the present consider the arms of the 
 lever as straight and uniform, or approximately so. 
 
 90. Levers are sometimes divided into three classes ac- 
 cording to the relative position of the points where the power 
 and the weight are applied with respect to the fulcrum. 
 
 Thus in levers of the first class, rig. i. 
 
 the power and the weight are applied ^ j?| 
 
 on opposite sides of the fulcrum G, 
 but act in the same direction, as in ' 
 fig. 1. 
 
 In levers of the second class, Kg. 2. 
 
 the power and weight are applied . ■ 
 
 on the same side of the fulcrum, t f K 
 
 but act in opposite directions (as Yht 
 
 in fig. 2, the power being applied 
 at a greater distance from the Mcrum than the weight is. 
 
 In levers of the third class ^'^'^ 
 
 (fig. 3), the power and the weight J^^ 
 act on the same side of the fulcrum 
 
 '4 
 
 "p 
 
 =2^ 
 
108 OF THE MECHANICAL POWEES. 
 
 ia opposite directions, tlie power being nearer the fulcrum 
 than the weight is. 
 
 The second and third classes it will Tdc observed do not 
 substantially differ from each other in their character. 
 
 When a lever is employed practically to transmit force 
 applied at one point to some other point, — as, for example, 
 when a crowbar is employed to raise a block of stone — the 
 pressure applied by the hand to one end of the bar corre- 
 sponds to the power P in the above explanation, and the 
 pressure which the block exerts upon the other end of the 
 crowbar corresponds to the weight W, — the fulcrum being 
 the fixed obstacle against which the crowbar rests, and about 
 which it can turn if P and W do not balance each other. 
 
 We have familiar examples of the first species of lever 
 in the common steelyard, a poher, the hraJce of a pump, the 
 common claw-hammer ; — a pair of scissors and carpenter's 
 pincers are double levers of this kind, the joint being the 
 fulcrum. 
 
 An car, a cork-squeezer, a pair of nutcrackers are exam- 
 ples of the second class. In the case of the oar, the blade of 
 the oar in the water is the fulcrum. 
 
 The treadle attached to the axle of the wheel of a lathe, a 
 pair of shears, — are instances of the third class of levers, and 
 to this class we may refer the hones of the arm and fingers 
 when put in motion by muscular action. 
 
 90. Conditions of equilibrium of a lever. 
 
 (I) When the lever is a straight one and the power and 
 weight act perpendicularly to the arms, as in any of the three 
 cases represented in figs. 1, 2, 3 (Art. 90). 
 
 Let B be the force (or reaction) which the fulcrum exerts 
 upon the lever, and the lever upon the fulcrum in the opposite 
 
THE LEVEK. 109 
 
 direction, then the lever ABG is kept in equilibrium by the 
 three forces P, W, B acting at A, B, respectively, and 
 these forces must satisfy the conditions of equilibrium of three 
 forces (Art. 45). 
 
 Hence, since the directions of P and W are parallel, R 
 must also act in a parallel direction, and in 
 fig. 1. R = P+W, 
 fig. 2. B=W-P, 
 fig. 3. R = P-W. 
 Also the moments of any two of the forces P, W, R about a 
 point in the line of action of the third must be equal in mag- 
 nitude and of opposite tendency. Hence taking the moments 
 of P and W about C, we have P.AG=W.BG (i) in each 
 of the three cases. 
 
 In levers of the first class it is obvious from equation 
 (i) that P will be > or < TF according as ^C is < or > BC, 
 i. e. according as the fulcrum is nearer to P or to W. 
 
 In levers of the second class, P is always < W. 
 
 In levers of the third class, P is always > W. 
 
 (II) WTien the lever is of any form, and the power and 
 weight act in any given directions (fig. Art. 89). 
 
 In this case also the three forces P,.W,R must act in one 
 plane (Art. 45), and, taking moments about the fulcrum 0, 
 
 we get P.Op=Q. Oq (ii), 
 
 Op, Oq being the perpendiculars from the fulcrum upon the 
 lines of action of P and W {Q and W having the same 
 meaning). 
 
 The results (i) and (ii) may be stated thus : " the power 
 and the weight which balance each other on a lever must be 
 
110 OF THE MECHANICAI. POWERS. 
 
 inversely proportional to the lengths of the perpendiculars 
 drawn from the fulcrum upon their directions," or 
 
 P _ perpendicular upon direction of W 
 W~ perpendicular upon direction of P 
 
 92. The magnitude and direction of the pressure R which 
 the fulcrum exerts in case (ii) may be expressed thus, — 
 supposing for simplicity A OB to be a straight line ; 
 
 put CAO=^a, CB0 = I3, COB=0, AO^a, BO = l; 
 then resolving the three forces P, 
 W, R parallel and perpendicular 
 to AB, we get (^ -^X-^'i [fN !fi\n 
 
 Rcos9 = Pcosa— Wcos /3, 
 
 5sin^ = Psina+ PT sin/3; 
 whence, squaring and adding, 
 
 R = V{P' +W'- 2PWcos (a + /3)}, 
 also dividing the latter by the former 
 
 . „ Psina+ Ws'mS 
 Pcosa— M/ cos/3 
 
 which two equations express R and 9 in terms of known 
 quantities. 
 
 Ois. We might, in cases (I) and (II) , have obtained other 
 equations of condition by taking moments about some other 
 point; as, for example, about A, in which case we get 
 
 R.AC=W.AB, fig. Art. 90, 
 
 or, in fig. Art. 92, if Ar, Aeo be perpendiculars drawn from A 
 upon the lines of action of R, W, 
 
 R.Ar=W. Aco, 
 
THE LEVEE. 
 
 Ill 
 
 whicli gives B at once independent of P; but conditions so 
 obtained are not independent of those abeady obtained, but 
 might have been deduced from them, as the student will infer 
 by examining this case in particular, or by referring to the 
 more general case discussed in Art. 45. 
 
 93. If two weights balance each other on a straight lever 
 in any position which is not vertical, they will balance in any 
 other position of the lever. 
 
 Let P, Q be the two weights suspended from the points 
 A,Boi the lever whose fulcrum is C and centre of gravity Q-, 
 
 W— weight of the lever, draw HO horizontal in the vertical 
 plane in which the lever can move. Suppose the lever to be 
 in equilibrium when inclined at an ^ d to the horizon, the 
 points A, O, C, B being in a straight line, — then since P, Q, W 
 act in vertical lines, the reaction R of the fulcrum must also 
 be vertical, and we must have 
 
 B = P+Q+W. 
 
 .(i). 
 
 Also taking moments about the fulcrum C, we must have 
 P .AC cos 0+W.CGco3e=Q. BO cos (li). 
 
112 
 
 OF THK MECHANICAL POWEES. 
 
 or since 6 is not = 90°, and /. cos 6 is not = 0, we may divide 
 out cos 0, and obtain 
 
 P.AG+ W. Oa= Q.BG (iii) 
 
 as the condition of equilibrium — and this is satisfied if the 
 lever assume any other position A' O'B' inclined at any other 
 angle to the horizon. Hence the lever will balance in any 
 other position. 
 
 NoU. li 6 = 90° then cos ^ = 0, and we should not be jus- 
 tified in deriving equation (iii) from (ii) by dividing out cos 6, 
 in fact when the lever is vertical it will balance with any 
 weights suspended at A,B. — It is necessary that A,B, G and 
 Q — the point where the fulcrum acts on the lever — should be 
 in a straight ling. 
 
 94. The various kinds of balances which are in use for 
 determining the weight of substances are constructed on the 
 principle of the lever. We will here give a description of 
 the common or Roman steelyard, the Danish steelyard, and of 
 the common halance; referring the student for a more complete 
 account to Delaunay's Cours iUmentaire de Micanique. 
 
 The common or Boman steelyard. 
 
 Thisbalance consists of a straight 
 lever AB suspended by the point G, jj 
 
 and capable of turning about this ===#== 
 point. At a point A on the short -^* 
 
 arm is attached a hook (or some- 
 times a scale-pan), from which is 
 suspended the substance whose, weight W is required. A 
 ring D, carrying a weight P of constant magnitude, can 
 slide along the graduated arm GB till P and W balance 
 
THE STEELYARD. 113 
 
 each other about C, when the lever is horizontal. The 
 graduation at which P rests when this is the case indicates 
 the weight of the substance. 
 
 In graduating the arm BG account must be taken of the 
 weight of the lever: let Q be the weight of the lever, and 
 G its centre of gravity, D the point from which P is sus- 
 pended when it balances W sXA; then taking moments about 
 G, we have 
 
 P. GD+Q.aG= W. GA (a). 
 
 If on the arm GA we take a fixed point such that 
 P . GO = Q . GO, the equation (a) becomes 
 
 P.GD + P.GO= W.GA, or P.On^W. GA; 
 
 W 
 .-. 0D = ~. GA. 
 
 We may now graduate OB by taking distances from 
 successively equal to GA, 2GA, 3GA, ... and marking them 
 1, 2, 3, ... — if necessary these distances may be subdivided. 
 
 Suppose, for example, that P rests at the fifth graduation, 
 then 0I) = 5. GA, and .-. W= 5P, and the weight of P 
 being known that of W is known also. 
 
 Obs. By increasing GA, or by diminishing P, the sen- 
 sibility of the steelyard would be increased; i.e. the distance 
 would be increased between the points from which P must 
 be suspended in ord^r successively to balance two weights 
 of given difierence. 
 
 For suppose B' the point of suspension of P when the 
 weight is W; 
 
 then P.0B'=W'. GA, 
 
 and P.0B = W. GA; 
 
 p. M. 8 
 
114 OP THE MECHANICAL PQ-WEES. 
 
 therefore P.BD' = OA. {W - W), 
 
 HA 
 or Dn=^.{W'-W); 
 
 1. e. W— W being given, DD' would be increased by an 
 increase of GA or by a diminution of P. 
 
 CA 
 We may regard - - as a measure of the sensibility of the 
 
 steelyard, and this being constant in the same steelyard for 
 diiferent positions of D, we infer that the same steelyard is 
 equally sensible for all positions of P. 
 
 The name of this steelyard has often led to a mistaken 
 idea of its origin — Romman is an Eastern word for the pome- 
 granate, and the form of the weight P gave rise to the name. 
 
 95. The Danish Steelyard. 
 
 This steelyard consists of a bar 
 AB terminating in a ball B which A ••• • ^ - ■ ~% ^ 
 
 serves as the power, and the sub- ^ ' Jf 
 
 stance to be weighed is suspended 
 
 from the end A; the fulcrum C — which is frequently a loop 
 at the extremity of a string by which the instrument is 
 suspended — is moved backward or forward till P and W 
 balance about it. 
 
 To graduate the Danish Steelyard. 
 
 Let P be the weight of the bar and ball of the steelyard, 
 which we may suppose to act through its centre of gravity G: 
 and let C be the position of the fulcrum when the substance 
 
THE COMMON BALANCE. 
 
 115 
 
 whose weight is W balances P about the fulcrum. Taking 
 moments about C, we have 
 
 P.CG = W.AO = P.{AG-AO); 
 
 P. AG 
 
 AC = 
 
 P+W 
 
 W; 
 
 by making TF successively equal to P, 2P, 3P...the successive 
 graduations are determined. 
 
 CoE. It is obvious from the formula (a) that the distances 
 of successive graduations from A are in harmonic progres- 
 sion. 
 
 96. The common halance. 
 
 This balance consists of a lever AB called the beam, 
 suspended from a fulcrum C about 
 which it can turn freely; the 
 point is a, little above the centre 
 of gravity G of the beam, and 
 from the extremities A, B of the 
 arms GA, GB (which ought to be 
 similar and equal) are suspended 
 two scale-pans, in one of which is 
 placed the substance whose weight 
 W is required, and weights of 
 known magnitude are placed in 
 the other till their sum P just balances W; this being the 
 case if the beam be exactly horizontal in a position of 
 rest. In this case if the arms are perfectly equal and 
 similar, and the scale-pans also of equal weight, P will be 
 exactly equal to W, If these weights differ by ever so little, 
 
 «— 2 
 
116 or THE MECHANICAL POWERS. 
 
 the horizontality of the beam will be disturbed, and after 
 oscillating for a short time, it will rest in a position inclined 
 to the horizon, and the greater this inclination is for a given 
 difference of P and W the greater is the sensibility of the 
 balance. A simple way of testing the accuracy of a balance 
 is by interchanging P and W in the scales. The balance 
 ought to retain the same position when this is done. 
 
 97. To determine the position of equilibrium of a balance 
 when loaded with unequal weights. 
 
 Let P and W be the weights in the scales. AB = 2a ; 
 h = the distance of G the fulcrum from the line joining A, B. 
 W the weight of the beam and scales, and k the distance 
 from (measured along the line h) of the point through which 
 the resultant of W acts — k remains unchanged when the 
 balance is tilted, — 6 the angle which the beam makes with 
 the horizon when there is equilibrium. 
 
 If we take moments about C, the algebraic sum must be 
 equal to zero. 
 
 Now the perpendicular from 
 
 on the direction of P= a cos ^ — A sin ^ ; 
 
 W=acoBd + hsm6; 
 
 W' = k&md; 
 
 we shall have then, taking account of the tendency of the 
 moments of the several forces, 
 P{acoae-h sin 0) - W (a cos ^ + A sin 6) - W'k sin ^ = ; 
 
 '■^^"^^--{P^rwyh^rwlc- 
 
 This equation determines the position of equilibrium. 
 
THE COMMON BALANCE. 117 
 
 98. The requisites for a good balance are 
 
 (i) The beam ought to be horizontal when loaded with 
 equal weights in the scales at A and B. This will be the 
 case if the scales are of equal weight, and if the line drawn 
 through G at right angles to AB divides the beam into two 
 similar and equal arms. 
 
 (ii) The balance ought to be sensible; i. e. the angle 
 which the beam makes with the horizon ought to be easily 
 perceptible when the weights F and W suspended at A and 
 B differ by a very small quantity ; and the greater tan 9 is 
 for a given small difference P — W, the greater is the sensi- 
 bility of the balance. We may take p — rr^ as a measure of 
 the sensibility, and hence we see that this requisite will be 
 secured by making as small as possible; 
 
 thus the smaller h and k are made, the greater will be the 
 , sensibility of the balance. 
 
 (iii) The balance ought to be stable; i.e. if the equi- 
 librium be a little disturbed either way, there ought to be a 
 decided tendency to return to its original position of rest. 
 This tendency, for any position of the beam, will be measured 
 by the moment q{ the forces tending to restore the beam to 
 its former position of re.st. If for example P=W, then when 
 the beam is inclined at ^ ^ to the horizon the moment of the 
 forces which tend to diminish 0, and therefore to restore 
 the balance to its position of equilibrium, is 
 
 {iP+W)h+W'Ic] sine. 
 
118 OF THE MECHANICAL POWEES. 
 
 Hence this ought to be made as large as possible in order to 
 secure the third requisite. 
 
 This condition, it will be observed, is to some extent at 
 variance with the condition for sensibility ; but they may be 
 reconciled by making (P+ W) h + W'k considerable and a 
 large; i.e. by placing the fulcrum at some distance above 
 the centre of gravity of the beam, and by making the arms 
 long. 
 
 In a balance of great delicacy the fulcrum should be as 
 thin as possible — it is generally a knife-edge of hardened steel 
 working upon agate plates. 
 
 The comparative importance of these qualities of sensi- 
 bility and stability in a balance will depend upon the service 
 for which it is intended : — for weighing heavy goods, stability 
 is of more importance ; — the balance employed in a chemical 
 laboratory must possess great sensibility, and such instru- 
 ments have been constructed to indicate a variation of weight 
 as small as a million-th part of the whole, — and even less. 
 
 99. There are various kinds of compound balances 
 formed by combinations of levers in use for weighing heavy 
 articles, as merchandise, baggage, &c. — it will suflSce here to 
 give a brief description of the arrangement of the levers in the 
 Balance of Quintens in a simple form. 
 
 The figure represents a section of the machine by a plane 
 dividing it into two symmetrical parts. 
 
 The platform AB upon which the weight Q is placed is 
 supported at one end upon the knife-edge fulcrum E, and at 
 
QUINTENZ' BALANCE. 
 
 119 
 
 the other hj a piece DH -which is connected with 1;he upright 
 piece BChj & strong brace CB. 
 
 flsiRJ 
 
 [te 
 
 he: 
 
 GF is a lever turning about a fulcrum F and connected 
 with the horizontal lever LMN by a vertical rod GL ; HK is 
 another vertical rod connecting DH with the lever LMN 
 which turns about the fulcrum M, and from the end N of this 
 lever the scale-pan P is suspended. 
 
 The ratio of FE : FO is by construction the same as the 
 ratio KM : LM, — usually 1 : 5. 
 
 The weight Q thus produces pressures at E and H: the 
 pressure at E by means of the lever FG and rod GL trans- 
 mits a pressure to the lever LMN at L, and the pressure at H 
 is transmitted to the same lever LMN at K, — and in conse- 
 quence of the ratios FE : FG and KM : LM being equal, the 
 pressure at L produces the same effect on the lever LMN 
 as a pressure equal to that at E would do if applied at K 
 
 Thus the effect on the lever LMN is the same as if the 
 whole weight Q were suspended at K, and equilibrium is 
 produced by placing suitable weights in the scale-pan P. 
 
 The ratio KM : MN is commonly 1 : 10, — so that the 
 weight of Q is ten times that required to balance it at P. 
 
120 
 
 OF THE MECHANICAL POWEES. 
 
 100. The Wheel and Axle. 
 
 This machine consists of a cylin- 
 der SH', called the axle, and a wheel 
 AB, the two having a common axis 
 terminating in pivots 0, C, about 
 which the machine can turn; — the 
 pivots resting in fixed sockets at 
 0, C. A rope, to one end of which 
 the weight W is attached, passes 
 round the axle, and has its other 
 end fixed to the axle. Another rope 
 passes round the wheel, being at- 
 tached at one end to the circum- 
 ference of the wheel, and at the 
 other end the power P is applied. 
 The ropes pass round the wheel and 
 the axle in opposite directions, and 
 thus tend to turn the machine in 
 opposite directions. 
 
 Conditions of eguilihrium of the wheel and axle. 
 
 The efforts which P or TF" make to turn the machine 
 about its axis will be the same in whatever plane they act 
 perpendicular to the axis. 
 
 Let fig. 2 represent a section of the machine perpendicular 
 to its axis ; M and N the points at which the strings quit 
 the circumferences of the wheel and axle; join OM, ON, 
 which will be perpendicular to MP, NW respectively. 
 
 We may regard MON as a lever kept in equilibrium 
 about the fulcrum by the forces P, W acting at arms 
 
THE WHEEL AND AXLE, 121 
 
 MO, ON, and there will be equilibrium if P. MO = W. NO, 
 
 P NO . . 
 or ^= Y¥ri) !• c- if the power is to the weight as the radius 
 
 of the axle is to the radius of the wheel. 
 
 101. Ohs. If the thickness of the ropes cannot be neg- 
 lected, we must suppose the action of P and W to be trans- 
 mitted along the middle or axis of the ropes, and in this case 
 
 Oilf= radius of wheel + radius of rope, 
 0N= radius of axle + radius of rope. 
 Instead of the wheel AB (fig. 1), the power P is some- 
 times applied to a rigid rod fixed into the axle at right angles 
 to it ; and in the previous condition of equilibrium we must 
 take 0M= length of the arm at which P is applied. The 
 capstan is an example of this construction. 
 
 COE. 1. In a combination of wheels and axles, in which 
 the string passing round one axle also passes round the wheel 
 of the next machine, and so on, we should readily obtain 
 
 P _ product of radii of all the axles 
 W product of radii of all the wheels ' 
 
 Combinations of toothed-wheels are substantially examples 
 of this kind. 
 
 CoE. 2. If \\i& power and weigJit act in parallel directions 
 on the wheel and axle, and on opposite sides of the axis, the 
 pressure on the axis = P + W; but if they act on the same 
 side of the axis, the pressure on the axis = P~ 1^. (Art. 91.) 
 
 102. The Pully. 
 
 The pully is a small circular disc or wheel having a 
 uniform groove cut on its outer edge, and it can turn about 
 
122 
 
 OF THE MECHANICAL POWERS. 
 
 an axis which passes through its centre. This axis rests in 
 
 sockets within the block to which the 
 
 puUy is attached. When the block is 
 
 fixed, the pully is said to he fixed; 
 
 in other cases it is moveable. A cord 
 
 passes round the pully along the 
 
 groove, and at its extremities the 
 
 power and weight are applied. 
 
 The pully is very useful for changing 
 the direction of the tension of a string; 
 and as we shall here suppose the groove 
 to be perfectly smooth, the tension at 
 all points of the string between the points of application of 
 P and W will be the same. (Art. 43.) 
 
 In the following account of some of the more simple com- 
 binations of puUies, we shall neglect the weight of the strings, 
 and suppose the radius of any pully to be the distance from 
 the axis to the centre of the chord which passes round it. 
 
 103. Conditions of equilibrium on a single moveable pully. 
 (i) When the strings are parallel. 
 
 Since the tension of the string PAJBC -which, passes round 
 the pully is the same throughout, the tension 
 iipwards of the portions AP, BG are each equal 
 to P; and since there is equilibrium we may 
 suppose the strings AP, BG attached to the 
 pully at A and B, the points where they quit 
 the pully; and the weight W, which is sus- 
 pended from 0, the axis of the pully, is sup- 
 ported by the upward tension of the strings 
 AP, BG. Hence, considering A OB as a lever kept in equi- 
 
THE SINGLE FULLY. 
 
 123 
 
 libiium about a fulcrum 0, we have (Art. 90) 2P=W the. 
 condition required. 
 
 (ii) When the strings are not parallel. 
 
 Let tlie string quit the puUy at A and B. Then since 
 the tension along AP is equal to that 
 along BG, their resultant will bisect 
 the angle between them, and this re- 
 sultant must be equal and opposite to 
 the weight W suspended from the axis 
 of the puUy, and acting in a vertical 
 direction. 
 
 Hence AP, BG must be equally 
 inclined to the vertical; let 6 be this inclination, then the 
 resultant of the two tensions, which we may regard as acting 
 at A and B, is = 2Pcos^, and this must be equal to W; 
 i. e. 2Pcos 6 = W, — the condition of equilibrium. 
 
 If the weight of the puUy be taken into account, 
 let it be w, and we shall obtain 
 2PcQs6= W+w for the condition of 
 equilibrium. 
 
 If instead of a weight W hanging 
 vertically, a force R be applied to the 
 pully in direction OR by a string or 
 otherwise, we may shew as before that 
 when there is equilibrium AP, BG 
 must be equally inclined to the direc- 
 tion of R, and we shall have 2Pcos = i? for the condition of 
 equilibrium, 2d being the angle which AP makes with BG. 
 
 104. Conditions of eqmlihrium in a system ofpullies. 
 I. In a system of puHies in which the string which passes 
 
124 
 
 OF THE MECHANICAL POWEES. 
 
 round any puUy has one extremity fixed 
 and the other attached to the puUy next 
 above it (as in the figure), the portions not 
 in contact with any pully being all parallel. 
 Let <j be the tension of the string which 
 passes round the lowest pully, t^, t^,... 
 that of the string passing round the second, 
 third ...'pvilly, and let w^, w^, w^...'be the 
 weights of the pullies A^, A^, A^,.. 
 
 Then for the equilibrium of the pully 
 -4,, we shall have the equation of condition 
 
 2t,= W+w^ (1) (Art.91), 
 
 and for the equilibrium of A^, the force upon it downwards is 
 equal to the tension of string A^A^ + w^, and the force up- 
 wards is 2«j, hence we get 
 
 2t^=t^ + w, (2), 
 
 similarly, 2f^= t^ + w, (3), 
 
 and so on for every moueaSZe, pully; if there be n moveable 
 pullies the last equation will be 
 
 Now if we multiply the equations (1), (2), (3) ... (n) by 
 1.2.2"... 2""' severally, add the corresponding sides together 
 and strike out terms which cancel each other on opposite 
 sides of the resulting equation, we get 
 
 2X= W+ w, + 210, + 2\, + ... + 2"-*w„, 
 and it is clear that t„ = P, hence' the condition of equilibrium 
 becomes 
 
 2"P= TF+w>j + 2w,+ 2X+... + 2"-X (a). 
 
 Cor. 1. If the weight of the pullies be neglected, 
 
SYSTEM OF PULLIES. 
 
 125 
 
 and the condition (a) becomes 
 
 2"P= W. 
 
 COE. 2. If the puUies are all equal and the weight of 
 each = w, the condition (a) becomes 
 
 2"P= W+ (1 + 2 + ...+ 2"-') 10, 
 or 2''P=W+{r-l)w, 
 
 which may be written 
 
 2''(P-M?)= TT-w. 
 
 105. II. In a system of pullies where there are two 
 blocks, and the same string passes round 
 all the pullies (as in the figure), the parts 
 of the string between successive pullies 
 being parallel. 
 
 Since the tension of the string is the 
 same throughout^ if n be the number of 
 strings at the lower block, nP will be the 
 resultant upward tension of the strings 
 upon the lower block, and this must be 
 equal to W when there is equilibrium, 
 that is, nP= W is the condition required; 
 the weight W including the weight of the 
 lower block. 
 
 106. III. In a system of pullies 
 where the string which passes rotind any 
 pully is attached at one end of it to the 
 weight, and at the other end to the next 
 pully (as in the figure), the strings being all parallel. 
 
126 
 
 OF THE MECHANICAIi POWERS. 
 
 Let «,, ij, «3... be the tensions of the 
 strings which pass round the successive 
 pullies A^, A^, A^..., and let w^, w^, w^... 
 be the weights of these pullies severally. 
 Then for the equilibrium of the pullies 
 A A , A ...in succession, we shall have 
 
 ^ 
 
 
 <n=2«„_l + «'a.i 
 
 •w, 
 
 nhi. 
 
 
 n being the number of pullies of which « — 1 only are 
 moveaile. 
 
 Also for the equilibrium of T^we shall have W= resultant 
 of the upward tensions of the strings attached to the bar AD ; 
 
 i.e. t^+t,+ t, + ... + t„= W, (^). 
 
 Multiplying the n equations of (a) by 2", 2""', 2"^ 2 in 
 
 order, and adding, we get 
 
 2«„=2w^^ + 2X.,+... + 2"-y+2"P. (7). 
 
 Again, adding equations (a), we get by means of (/S) 
 
 F'=P+Wi + Wj+... + w^, + 2(Tr-0; 
 
 i.e. 2«„= TF+P + w^ + Wj, + ... + w^, (8). 
 
 Subtracting (S) from (7), we get 
 
 W= (2"- 1)P+ (2":'- 1) Wj+ (2"-=- 1) w,+ ... 
 
 ...-fr{f-l)w^ + w^^, 
 
 the relation which must hold good between P and IF when 
 the system is in equilibrium. 
 
SYSTEM OP PULLIES. 127 
 
 COE. 1. If the weight of the puUies be neglected, 
 
 and the relation between P and W requisite for equilibrium 
 becomes 
 
 TF=(2"-1)P, 
 
 a result which the student may investigate independently. 
 
 Ohs. It is readily seen that the weight of the puUies 
 assists the power P in system III., but in the systems I, and 
 II. it increases W, 
 
 CoE. 2. If the weight W be suspended from a horizontal 
 bar AD, the point K to which W is attached must be such 
 that the resultant of the tensions at ^, -B, 0... passes through 
 K, otherwise the bar would not remain horizontal. 
 
 For example, suppose there are four puUies of equal 
 radius a : then the tension at A = P, at B= 2P, at C= 2"^, 
 and at D=2^P, the weights of the puUies being neglected; 
 then, taking moments about A, we must have K such that 
 (P+ 2P+ 2'P+ 23P) AK = P. + 2 . P. a + 2=P. 2a + 2^P. 3a ; 
 ,^ 2 + 2'. 2 + 2=. 3 34 
 
 ^•^•^^= l + 2 + 2-+2- "='r5^- 
 
 107. A very simple and useful combination of puUies is 
 employed in the Spanish Barton, the principle 
 of which will be obvious from the annexed 
 figure. C is a fixed puUy round which passes 
 a string attached to the two moveable puUies 
 A and B. The weight W is attached to B, and 
 the string which passes round A and B is fixed 
 at one end at D, and the power P acts at the 
 other end. 
 
 For the conditions of equilibrium we have 
 
128 OF THE MECHANICAL POWEES. 
 
 from the equilibrium of the puUy A, tension of string 
 ACB=2P+A= T suppose, and the tension of the string 
 PABD being the same throughout and equal to P, we have 
 for the equilibrium of puUy B, 
 
 B+W=2P-\-T=.\4.P+A; 
 .•.iP = W+B-A. 
 
 108. The Inclined Plane, 
 
 By an inclined plane, as a mechanical power, is meant a 
 plane inclined to the horizon, and the inclination is measured 
 as in Euclid, Book xi.. Definition 6. Or thus: If a vertical 
 plane be drawn perpendicular to the inclined plane, — which 
 for simplicity of definition we shall call a principal j)lane — 
 the angle between the lines of intersection of this vertical 
 plane with the inclined plane and a horizontal plane is the 
 inclination of the proposed plane to the horizon. 
 
 Conditions of equilibrium on an inclined plane. 
 
 When a body whose weight is W 
 is supported on an inclined plane hy 
 a force P, the direction of which makes 
 an angle e with the plane — the plane 
 being smooth. 
 
 I. Let the figure represent a 
 section of the inclined plane, made 
 
 by a vertical plane perpendicular to the inclined plane; 
 BAG= a, the inclination of the plane. Then the forces acting 
 on the body at are W its weight vertically, R the reaction 
 of the plane acting at right angles to the plane, and P the 
 given force. Hence, in order that the body may be in equi- 
 librium, P must act in the same plane with W and It, i. e. in 
 the vertical plane perpendicular to the inclined plane (repre- 
 sented in the figure). 
 
THE INCLINED PLANE. 129 
 
 Let FOB = e. Tlien resolving the forces which act on 0, 
 along the plane and at right angles to it, we get 
 
 Pcose — TF'sina = (i), 
 
 Psine + i? — PF'cosa=0 (ii). 
 
 Equation (i) gives the relation whicl\ must hold between 
 TFand P; and the second gives 
 
 7? ITT- D • -nrf sinasine"\ TFcosfe + a) 
 
 it = Tr cosa— Psme = PF cosa 1 = ^^ -' , 
 
 \ cos e / cos e ■ 
 
 which gives the pressure on the plane in terms of W. 
 
 109. II. Let the plane be rough /'audi first, let P act in 
 the vertical plane which is perpen- 
 dicular to the inclined plane— i. e. in a b\ 
 principal plane — (as in case I.). i2, 
 P, W, a, e, the same as in case I.: /a 
 fi^ the coefficient of the friction actu- 
 ally exerted, down the plane suppose, 
 so that /ij-B is the friction; then resolving' the "forces act- 
 ing on the body, parallel and at right angles to the plane, 
 we get 
 
 Pcos e — Wsm a — fi^R = 0, 
 
 Psine+B— Wcos a = 0, 
 whence we get 
 
 p_ jy^ sipg + Z^iCosa ,. 
 
 cos e -+■ /ij sin e ^ '' 
 
 B=W-^^^^^^±4- (ii). 
 
 cos € + fij sine ^ ' 
 
 Equation (i) gives the relation between P and W; and 
 (ii) the pressure on the plane. 
 
 If the friction acts up the plane, we have only to change 
 the sign of fi^ in the preceding iavestigation. 
 
 P.M. 9 
 
130 
 
 OP THE MECHANICAL POWERS. 
 
 Obs. If fi be the coefficient of maximum friction between 
 the substance of which the body is composed and the plane 
 (which is determined by experiment, and generally given in 
 tables of the coefficients of friction), fi^ cannot be greater than 
 fi numerically, and may be positive or negaltive so far as equi- 
 librium is concerned. 
 
 If the body is just on the point of moving wp the plane 
 
 , „ T;^sina + /icosa 
 
 u, = u and P = W -^ — -. — . 
 
 '^^ '^ cos e + /* sin e 
 
 If it be on the point of moving down the plane fi^ = — /jl and 
 „ TT^ sin a — u. cos a 
 cos e — /i sm e 
 
 If P have any value intermediate to these two, the body 
 will be in equilibrium, and for a given value of P the co- 
 efficient of the friction actually in operation will be given 
 , ... . Pcose— TFsina 
 
 ^yW5 ^•e-/^t=prcosa-Psine- 
 
 Secondly. If the direction of P does not lie in the vertical 
 plane which is perpendicular to the 
 inclined plane. 
 
 Let OD be the projection of OP 
 on the inclined plane, AB the sec- 
 tion of the inclined plane made by 
 the vertical plane through per- 
 pendicular to the inclined plane, 
 POD = e, BOD = p, BAC = a. 
 
 Now friction always acts in the direction opposite to that 
 in which the body would begin to move, if the friction were 
 to cease. 
 
 The forces acting on parallel to the plane are ,P cos e 
 along OD, and Wsin a along OA. 
 
THE INCLINED PLANE. 131 
 
 Hence jiB, the friction must be equal and opposite to the 
 resultant of these two forces. Let 6 be the angle which this 
 resultant makes with OA. 
 
 Then resolving the forces which act on (i) perpen- 
 dicular to the plane; (ii) along OA; and (iii) perpendicular 
 to OA along the inclined plane, we get successively 
 
 Psine + i2- TFcosa=0 (i), 
 
 Pcosecosp + fiBcoaO — Wsluo. = (ii), 
 
 Pcosesin/S — ijlBs\d.6 = (iii). 
 
 From these three equations we get P, R, and 6; i.e. the 
 force P necessary for equilibrium, the pressure on the plane, 
 and the directioii,in which friction acts. 
 
 If ;8 = 0, then = 0, and the results of this case coincide 
 with the preceding. 
 
 110. If /i = tan ^, the result of the first case of II. gives 
 
 p= ppsinjajf^ 
 cos {e — <f))' 
 
 and if we suppose e and P to vary so as to satisfy this relation, 
 we see that P is least when cos (e — ^) is greatest; i. e. when 
 e = ^, and the least force which will pull the body up the 
 plane is = TF'sin(a + 0). 
 
 Also the result P = W . z , . compared with that of 
 
 cos (e + 9) -^ 
 
 case I. shews that the' condition of .equilibrium on a rough 
 
 plane is the same as that on a smooth plane whose inclination 
 
 to the horizon is increased or diminished by the angle ^, the 
 
 direction of P remaining unchanged-r-imcreasec? or diminished 
 
 according as the friction acts down or wp the plane. 
 
 9—2 
 
132 
 
 OF THE MECHANICAL POWERS. 
 
 CoE, If the force P acts horizontally, then e = — a, and 
 the conditions of equilibrium become 
 
 in case I. P = TF tan a, 
 
 in the first instance of case II. P=W 
 
 ^ sin a.-\- fjb cos a 
 cos a — /i sin a 
 
 111. The screw. 
 
 The screw is a spiral thread running along the surface of 
 a circular cylinder, which may be imagined to be generated 
 thus: 
 
 Let ^ G' be a rectangle whose base AB is exactly equal to 
 the circumference of the cylinder ; make 
 the rectangles BD, OF, ER... equal in 
 every respect, and draw the straight lines 
 AG, BE, EG...; then if the rectangle 
 BH be applied to the surface of the 
 cylinder so that the base AB coincides 
 with the base of the cylinder, the broken 
 lines AG, BE, EG... will form a con- 
 tinuous line on the surface of the cylinder, 
 the point C coinciding with B, E with F, 
 and so on. If we now suppose this line 
 to become a protuberant thread, we obtain 
 a screw, in which the distance between any point of one 
 thread and the one next below it, measured parallel to the 
 axis of the cylinder, is everywhere the same and equal to 
 BG: — the angle CL45 which the thread at any point makes 
 with the base of the cylinder is called \h& pitch of the screw. 
 
 The screw formed on the solid cylinder, as above, works 
 .in a hollow cylinder of equal radius, in which a spiral groove 
 is cut exactly equal and similar to the thread on the solid 
 
THE SCEEW. 
 
 133 
 
 cylinder, and in which groove the thread of the former can 
 work freely. 
 
 , A solid and hollow screw related as above are called 
 companion screws ; and when in action, one of them is -fixed 
 and the other is turned by means of a lever fixed into the 
 cylinder at right angles to its axis. By turning the lever a 
 weight is raised, or a pressure produced, at the end of the 
 screw, which pressure acts in direction , of the axis of the 
 screw. 
 
 When the solid screw is small, it is sometimes called 
 a nut. 
 
 L 
 
 W 
 
 112. The figure in the margin 
 will convey some idea of one mode of 
 applying a screw to produce pressure, 
 and in all the various applications 
 of the screw we may regard a power 
 P as applied at the extremity of an 
 arm which is at right angles to the 
 axis of the screw so as to produce 
 a pressure in direction of the axis, 
 which we shall call the weight. 
 
 The form of the thread of the 
 screw is not always the same in 
 difierent screws ; we shall suppose a section of it made by a 
 plane" through the axis of the screw to be rectangular, so that 
 the surface of the thread will present the same appearance as 
 the under surface of a circular spiral staircase. . 
 
 113. Conditions of equilibrium on the screw. 
 
 In Investigating the relation between F and W we shall 
 for the sake of simplicity suppose the screw to be vertical. 
 
134 
 
 OF THE MECHANICAL POWERS. 
 
 ■^-^^P 
 
 and the pressure which balances P to be a weight W placed 
 on the end of the screw, and which the screw supports. 
 
 I. When the screw is smooth. 
 
 We may suppose the whole weight 
 W to be distributed along the surface 
 of the screw which is in contact with 
 the companion screw. 
 
 Let w be a portion of W supported 
 at by the pressure r of the companion 
 screw and by a force p acting hori- 
 zontally, i.e. perpendicular to the axis 
 of the screw: a = the pitch of the screw, 
 i. e. the complement of the angle which 
 the tangent to the thread of the screw at makes with the 
 axis of the screw. 
 
 Theu the conditions of equilibrium of w under the* action 
 of r and p are the same as those of a body resting on an 
 inclined plane (inclination = a) under the action of a force p 
 acting horizontally; hence ^ = w tana. (Art. 110, Cor.) 
 
 Similarly, if w, w" ... be the portions of the weight sup- 
 ported at successive points by the forces p\ p"... we should 
 have 
 
 ^'=M)'tana, p" = w"tana 
 
 and .•. p+p +p" + ... = {w + w +w" + ...) tana. 
 
 Now w + w' + w"+... must equal TT the whole weight, 
 and p, p', p". . . acting at the surface of the cylinder perpen- 
 dicular to its axis produce the same effect to turn the cylinder 
 round its axis as the power P acting at the arm GA. 
 
 Hence the moments oi p, p', p"... about the axis must 
 together be equal to the moment of F about the same ; i. e. if 
 
THE SCEEW. 135 
 
 c be the radius of the cylinder, a the length of the arm CA 
 at which P acts, we must have 
 
 2>c -k-ip^c-^-p'c + ... = Pa ; 
 
 whence we get — = W tan a ; 
 c 
 
 _ TFc tan a ,., 
 
 or P= (i); 
 
 the relation between P and 1^ req[uired. 
 
 Since 27rc tan « = distance between two threads, _measured 
 parallel to the axis, 
 
 and lira = circumference of the circle described by A' 
 
 we may write the condition (i) in the form 
 
 P _ 27rc tan a _ distance between two threads 
 W 27ra circumference of circle, radius CA ' 
 
 II. If the screw he rough. 
 
 In this case supposing W distributed, as in case I., the 
 forces which act at are the weight of w vertically, p hori- 
 zontally, p the normal pressure on the thread of the screw, and 
 fip the friction along the surface of the thread ; hence taking 
 the conditions of equilibrium on a rough inclined plane as in 
 (Art. 109), i. e. resolving the forces along the tangent line and 
 perpendicular to it, we get (supposing the friction to act down 
 the screw ; i. e. to oppose P,) 
 
 p cos a — w sin a — /*/) = 0, 
 
 ^ sin a + m; cos a — /} = ; 
 
 whence we obtain 
 
 sin a + u cos a w 
 
 p = w— : — ; p = 
 
 COS a — yii sm a ' cos a — /* sm a 
 
136 OF THE MECHANICAL POWERS. 
 
 Or if we pat /* = tan ^, these become 
 
 sin (a + d)) ' w cos d>. 
 
 ■^ ' COS t(a +^) ' ^ cos {oi. + 4>)' 
 whence as before 
 
 ^+p'+p"+ ...={w + w' + w"+ ...)ta,n(ai + ^), 
 
 and P- = W. tan (a + .jf)) ; or P=W- tan (a + <f>), 
 
 C €C 
 
 the relation between Pand W. 
 
 114. OJs. If the friction acts up the screw (i. e. assists P), 
 then we must change the sign of /* and therefore of j>, and 
 we get in this 'Case 
 
 P=TF-tan(a-<A). 
 
 Note. Sinjee the distance between two threads, measured 
 parallel to the axis, is the same at all points of the screw, but 
 the length of one revolution of the screw is greater at greater 
 distances from the axis, it is clear that the pitch a. is different 
 at points on the surface of the thread which are at different 
 distances from the axis, — being greatest at points nearest the 
 axis. But when the screw is smooth, as in case I., the 
 relation between P and W, viz. 
 
 P _ distance between two threads 
 W circumference of the circle, radius OA ' 
 depends only on the distance between the threads and the 
 length OA ; hence this result will be true whatever be the 
 ireadth of the thread. 
 
 115. The wedge. 
 
 The wedge is a solid triangular prism made of hard 
 material such as iron or steel, and is used for separating two 
 
THE WEDGE. 
 
 137 
 
 bodies, or two parts of the same body, which adhere power- 
 fully to each other. The edge of the wedge is introduced 
 between the parts of the substance, and it is then driven 
 forward by smart blows of a hammer applied at its back, or 
 by some equivalent process. Hatchets, chisels, nails, carpenters\ 
 planes, swords,— are modifications of the wedge. 
 
 The action of the wedge is so essentially dynamical that 
 it would serve no useful purpose to discuss its statical con- 
 dition at any great length; we will only obtain the condi- 
 tion of its equilibrium in a very simple case. 
 
 Condition of equilibrium of a wedge. 
 
 Suppose the wedge isosceles, and let 
 the figure represent the position of the 
 wedge inserted in the obstacle and in 
 contact with it at A and B; B, R' the 
 pressures perpendicular to the faces of 
 the wedge at B, A ; fiB, fiB' the friction 
 on the wedge at those points; W the 
 force applied at the middle point of the 
 back of the wedge, and a the angle between 
 the faces of the wedge; then resolving 
 the forces which act on the wedge in 
 direction of W which bisects the angle a, 
 and at right angles to this we get 
 
 W+fi{R + Ii')cosl-{M + B')Bml = 0, 
 
 lilB- E) sin 1 + (iS -5') cos| = 0, 
 
 the latter equation gives B= B', and subs,titu!ting ibis in the 
 former, we get 
 
138 OP THE MECHANICAI, POWERS. 
 
 W=2B Tsin- — ^cos|j, 
 
 ft ibeing the coefiSeient of friction actually in operation. 
 
 It may be remarked that in many cases the wedge is kept 
 in its place by friction alone, — in such cases W=0, and 
 
 .a a . . , a 
 
 .•. sm- — /tcos- = 0; i.e. /« = tan-, 
 
 which gives the coefficient of friction actually required for the 
 equilibrium of the wedge. 
 
 116. Principle of Virtual Velocities. 
 
 Def. If A be the point of application of a. force P, and 
 this point receive a small 
 displacement so as to come ■ 
 to A!, the small space AA' 
 is called the virtual velocity 
 of the point A, and if A'a 
 be drawn perpendicular to 
 AP, the small space Aa is called the virtual velocity of the 
 force P, and is regarded as positive or negative according as a 
 lies on the side of A towards which P acts, or the opposite, — 
 in other words, the virtual velocity or displacement of the 
 point of application resolved in direction of the force is the 
 virtual velocity or displacement of the force: — ^the direction of 
 the force A'P in the new position being supposed to remain 
 parallel to AP, or very nearly so. 
 
 If A'Aa = a = A'AP, we have Aa = AA! . cos a ; hence 
 the virtual velocity of a force is equal to the virtual velocity 
 of its point of application multiplied by the cosine of the 
 angle, which the direction of the displacement of the poiilt 
 makes with the direction of the force. 
 
VIETUAL VELOCITIES. 139 
 
 The product of any force into its virtual velocity is called 
 the virtiial moment of the force; 
 
 117. When a machine or system of bodies is in equilibrium 
 ttnder the action of several forces, if the point at which any 
 one force is applied be slightly displaced without breaking the 
 connexion of any of the parts of the system, the points at 
 which the other forces are applied "srill, in general, also be 
 displaced to an extent dependent upon the displacement of 
 the first point ; and the following singular relation exists 
 among the forces and their several displacements or virtual 
 velocities, viz. The algebraic sum of each force multiplied hy 
 its virtual velocity is equal to zero, or, iii other words, The 
 algebraic sum of the virtual moments of a system offerees in 
 equilibrium is zero. 
 
 This is sometimes called the equation of virtual velocities. 
 
 Since the displacement of the several points would all 
 take place in the same time, it is obvious that they would, if 
 small, be in the ratio of the velocities of the several points; 
 and further, since a system in equilibrium cannot move of 
 itself, the displacements above supposed are hypothetical or 
 virtual only, and such as would ensue upon the application of 
 some additional force which is supposed to cease as soon as- 
 the displacement is effected, and the system to be in equi- 
 librium and at rest in its new position. Hence the term 
 virtual velocity. 
 
 118. The proof of the principle of virtual velocities in 
 its general form is of too difficult a character to be introduced 
 into an elementary work. We will here shew that the principle 
 holds true in the case of the lever, the wheel and axle, the 
 several systems of pullies, the screw, and the inclined plane; 
 in other words, when a power and a weight balance each 
 
140 OF THE MECHANICAL POWEES. 
 
 other oil aiiy one of these macliines, if the power be slightly 
 displaced, the consequent displacement of the weight is such 
 that P. displacement of P= W. displacement of W, 
 or P. virtual velocity of P= W. virtual velocity of ^^...(1). ._ 
 
 In each of the following cases, the student will observe 
 that if the displacement of P is in direction of P's action, 
 that of W will be in the direction opposite to W's action; 
 i. e. the virtual velocities of P and W are of contrary alffebraia 
 sign, so that, although for the sake of obtaining more conve- 
 nient formulae we shall neglect the algebraic signs of the 
 displacements and regard their actual magnitude alone, the 
 equation (1) would algebraically be written 
 
 P . P's virtual velocity + W.. W's virtual velocity = 0. 
 
 119. Case I. When P and W balance each other on a 
 bent lever. 
 
 Let AGB be a bent lever whose fulcrum is C; CM, ON 
 the perpendiculars from G on 
 the lines of action of Pand W. 
 Let the lever turn about C 
 through a small angle A OA^ = a 
 so as to come into the position 
 A,OB/, then A A,, BB, are small 
 circular arcs which may approxi- 
 mately be regarded as straig'bt 
 lines, and the angles GAA', 
 GA^A, CBB,, GBfi as very nearly right angles. 
 
 Hence if Afl, B]b be drawn perpendicular to AP, BW, 
 then Aa, Bb will be the displacements or virtual velocities of 
 P and Irrespectively, and we shall have AA^ = a..AG, and 
 Aa = AA^ . cos A^Aa = AA^ . sin GAM 
 = a.AG- sin CAM= a. . GM. 
 
VIRTUAL VELOCITIES.- 
 
 141 
 
 Similarly, 
 
 Bh = BB, . cos B,Eb = BB, . sin CBN 
 = a. CB sin CBN ^ a. GN. 
 
 Whence ^«.^-f(A,..,89), 
 
 and .:P.Aa= W.Bb, 
 or P.P^a virtual velocity = W , TF's virtual velocity, 
 
 120. Case II. The wheel and axle. 
 
 Let the machine be turned about its axis through a small 
 angle a, so that the Wnt MON comes 
 into the position m On ; then Mm, Nn 
 will represent the lengths of string 
 which have unwrapped from the wheel 
 and wrapped upon the axle respec- 
 tively; i.e. Mm, Nn will be equal to 
 the corresponding displacements of P 
 and W. 
 
 Mm_ MO.^a _MO 
 Nn ~ NO.^a.~NO 
 
 And 
 
 NO.^a. 
 
 _W 
 ~ P' 
 
 .'. P. P's virtual velocity = W. TF's virttial velocity. 
 
 121. Casein. The single moveable ^ully, 
 when the strings are parallel. 
 
 If the weight be raised through any space 
 s, it is clear that the parts of the string on the K c 
 opposite sides of the puUy have to be shortened, 
 each by a length s, in order that the string 
 may become tight; i.e., P must move through a 
 space 2s; 
 
1-42 
 
 OP THE MECHANICAL POWERS. 
 
 P's virtual velocity _ 2s _ W 
 ' ' W's virtual velocity s P ' 
 
 P . P's virtual velocity = W. TF's virtual velocity. 
 
 122. CaselY. The single moveable pully when the strings 
 are not parallel. 
 
 Suppose the string fixed at B and to pass round a small 
 peg at A. Let tlie centre of the pully 
 be raised from to 0, in a vertical 
 direction, and let Q, Q^ be the points 
 where the string quits the pully in the 
 two positions; draw 0^^,,!r perpen- 
 dicular to AQ and therefore parallel to 
 OQ. The angle Q,0,Q,, is small and 
 equal to QAQ,, and we may regard 
 Q^ §,, as very nearly equal to the small 
 arc of the pully intercepted by O^Q^ and 0,Q,/, therefore 
 since ^ QOG= ^ Q„0,C„ we shall have QG= Q„Q,G„ and 
 AQi, = AT very nearly. 
 
 Hence it is clear that TQ will very nearly be the differ- 
 ence of lengths of the strings AQO and AQ^C/, and by the 
 raising of the pully BC is shortened just as much as ^C; 
 i. e. 2 TQ is the length of string which passes over A, — in other 
 words, 00, and 2TQ are the corresponding displacements of 
 PTandP; 
 
 P's virtual velocity 2TQ 
 W's virtual velocity ~ 00^ 
 
 = 2 cos 0, 
 if 26 be the angle between the strings, 
 
 _W 
 ~ P' 
 
 = 2 sin TO.O 
 
 and this is 
 
VIRTUAL VELOCITIES. 
 
 143 
 
 .'. P. P's virtual velocity = W. TF's virtual velocity/. 
 
 123. Case V. In the system of jaulUes described (Art. 
 104). 
 
 If the puUy A^^ to which W is fixed 
 be raised through a space s, the next 
 pully A^ will be raised through 2s ; the 
 next A^ through 2. 2s, i. e. 2\ and so on — 
 so that if n be the number of moveable 
 pullies, P will move through 2"s ; 
 
 ■P's virtual velocity _ 2"s _„n_^. 
 W'a virtual velocity s F ' 
 
 .'. P . P's virtual velocity = W, TF's virtual 
 velocity, 
 
 124, Case VI. In the system ofjoullies described (Art. 
 106). 
 
 If the weight be raised through a space 
 s, the pully A^ will be lowered through s ; 
 A^ will be lowered through 2s in conse- 
 quence of A^ being lowered through s, and 
 through 5 besides in consequence of W 
 being raised through s; i.e. A^ will be 
 lowered through (2 + 1 ) s ; similarly A^ will 
 be lowered through 2(2 + l)s + s; i.e. 
 through (2'' + 2 + l)s... and if there be n 
 pullies P will be lowered through a space 
 (2"-i + 2"-^+ ... + 1) s, or (2"- 1) s. 
 
 3 
 
 ■ 1 = 
 
 ^^^^^ P's virtual velocity ^ (2" - 1) « ^ „. 
 
 ' W'a virtual velocity s x- ■ 
 
 .•. P. P's virtual velocity = W. TT's virtual mlocity. 
 
144 
 
 OF THE MECHANICAL POWERS. 
 
 Note. In the last two cases the weight of the pullies has 
 been neglected, — we leave it as an exercise for the student to 
 prove that when the weight of the pullies is taken account of 
 virtual moment of P-\- that of W + that of each pully = 0, 
 
 125. Case VII. In the system of pullies described in 
 (Art. 105). 
 
 If the weight Ibe raised through a space s, each of the 
 strings at the lower block will be shortened by a length s, 
 and consequently P will have to move through ns in order that 
 the string may become tight, Hence 
 
 P's vu-tual velocity _ns _ _ W 
 W'si virtual velocity s ~ P ' 
 
 .•. P. P's virtual velocity = W. Ws virtual velocity. 
 
 Cor. In this system of pullies whilst a length s of string 
 passes round the pully A, a length 2s will 
 pass round the next pully B, 3s round the 
 next pully G, 4s round D, and so on. 
 
 If then the radii of the pullies^, B,0... 
 are in the ratio of consecutive numbers 
 1, 2, 3, ...the pullies will all revolve 
 through the same angle, since the arcs of 
 circles subtending equal angles at the 
 centre are proportional to the radii of the 
 circles. 
 
 Instead of supposing the pullies to be . 
 distinct and separate, we may suppose 
 circular grooves cut in the upper block (in 
 the figure) with radii in the proportion of 
 the even numbers 2, 4, 6, . . . and in the 
 
VIRTUAL VELOCITIES. 145 
 
 lower block grooves with radii in. the proportion of the odd 
 ijumbers 1, 3, 5, ... and these grooves will answer the purpose 
 of so many distinct puUies ; and every point of the circum- 
 ference of each groove moving just as fast as the part of the 
 string which is in contact with it, there will be no sliding or 
 rubbing of the' string over any groove. ■ This is the principle 
 of White's pully. 
 
 126. Case VIII. The Screw (Fig. Art. 113). 
 
 If the power P make a complete revolution, it is obvious 
 that the weight W will be raised through a space equal to 
 the distance of two threads measured parallel to the axis of 
 the screw, and proportionately for any smaller displacement 
 of P. Hence 
 
 P's virtual velocity _ circumference of circle described by P 
 TF's virtual velocity distance between two threads 
 
 _W 
 
 ~ P' 
 
 ,'. P. P's virtual velocity = W. W'a virtual velocity. 
 
 127. Case IX. The inclined plane supposed smooth. 
 
 Let the weight be displaced along 
 the plane through a small space 00^, 
 the direction of P remaining appre- 
 ciably the same after as before the 
 displacement, draw Ojp perpendicular 
 to OP, and On horizontal; then Op 
 and O^n are the corresponding dis- 
 placements of P and W. Hence 
 
 P's virtual velocity _ Op _ 0^ cos e _ cos e _ W ^ 
 W's, virtual velocity ~ O^n ~ C Oj sin a sin a P ' 
 P.M. 10 
 
146 OF THE MECHANICAL POWEES. 
 
 .'. P. P''b virtual, vslacity = W. W^s virtual velocity. 
 
 Note. The student will have little difficulty in proving 
 the equation of virtual velocities to hold' good in the case of a 
 rough inclined plane. 
 
 128. Case X. Any comhination of ' Machines. 
 
 We have seen that the virtual moment of thepower P applied 
 to any machine is equal to the virtual moment of the weight or 
 force which balances P in that machine. If now we have any 
 comhination of machines A,B, C...the force which balances 
 P on -4 may be regarded as a power appKed to B, and the 
 force which balances this on B may be regarded" as a power 
 applied to 0, and so on; and from what precedes we infer that 
 the virtual moments of each of these forces are equal. If then 
 P be the power applied to the first of a train of machines, and 
 W be the weight or force which balances it on the last machine 
 of the train, we shall have 
 
 P. P's virtual velocity = W. PF's virtual velocity. 
 
 Conversely, if this equation is satisfied in any combination 
 of machines, we readily infer that P and W balance each 
 other. 
 
 129. Mechanical advantage ani efficiency. 
 
 Def. The mechanical advantage of a simple' machine is 
 
 the number expressing the multiple which the weight or force 
 
 produced is of th.& power or force producing it, — in other words, 
 
 . W 
 it is the ratio -^ ; for instance, in the case of the system, of 
 
 puUies (Art. 104) the mechanical advantage = -g = 2". 
 
 If for example there be four moveable puUies then the 
 
MECHANICAL ADVANTAGE. 147 
 
 mechanical advantage is 16, and a power equivalent^to 8lbs. 
 would be able to raise a weight of 16.8 or I28lbs. 
 
 The advantage, of a combination of majchines will be equal 
 to the product of the a(?T?awtet^es of the several machines in the 
 combination, 
 
 130. In the several machines described in this chapter we 
 have supposed the forces just to balance each other, so that no 
 motion would ensue; and we have also in most cases neglected 
 the friction which will in practice exist among the parts of the 
 machine. These suppositions, however, are not quite accurate 
 when any mechanical agent is emplbyied to produce a Certain 
 effect by means of a machine ; as for instance, when the pres- 
 sure of the air is employed by means of a windmill to ^nd 
 corn, or a horse draws a cart along a rough road horizontal or 
 inclined, or a locomotive is propelled along a railroad by steam 
 pressure. In all such cases it is obvious that the pressure 
 applied at first to put the machine in motion must exceed the 
 resistance to be overcome; and so long as this excess continues, 
 the rate of working of the machine will be increasing: when 
 this rate of working has become sufficiently great, if we 
 suppose the excess of the fbrce over the resistance to cease, the 
 machine will go on working uniformly, and the force or power 
 applied viiMjust balance the resistance. 
 
 131. The amount of work done by a machine is com- 
 monly measured- by the product of the pressure exerted at the 
 work and the space through which it is exerted,, no regard 
 being had to the rate or speed at which the machine is 
 working. 
 
 Def. This is sometimes called the Idbouring farce or work 
 done or, efficiency, — ^in other words, we may define the efficiency 
 
 10—2 
 
148 OP THE MECHANICAL POWEES. 
 
 of a force to be the product of the number of units of force ex-^ 
 erted into the number of units of space through which it acts. 
 Thus for illustration, if we take as the unit of efficiency or 
 the dynamical unit, the work performed in raising lib. Terti- 
 cally through 1 foot, then the efficiency required to raise a ton 
 through 1 yard would be = 6720. 
 
 131. The standard of efficiency or worh done assumed by 
 Watt and adopted generally by engineers is 33000 ZJs. raised 
 through \ foot per minute, — the agent working steadily. This 
 is called a horse-power, and the efficiency of steam-engines and 
 other machines is commonly expressed in terms of this unit. 
 Thus if a machine of H horse-power can raise P lbs. through 
 f feet in t minutes, we shall have these quantities connected 
 by the relation of Pf= 33000 H. t. 
 
 132. We have seen in the case of the simple machines, 
 or any combination of them, that if the system be put in 
 motion, then 
 
 P.P's displacement = W. TT's displacement. 
 
 This result shews us that however the application of a force 
 be modified or rendered more useful by the intervention of a 
 machine, yet no efficiency is, gained thereby; and further, the 
 same result put in the form 
 
 P's velocity _ W 
 
 W'& velocity" i^' 
 
 shews that in whatever proportion the intensity of a force he 
 increased by means of a machine, yet the space through which 
 the increased force will operate will be diminished in the in- 
 verse proportion as compared with the space through which 
 the force applied must operate. Thus for the sake of illustrar 
 tion, suppose a weight of 500lbs. is supported on a machine 
 
MECHANICAL ADVANTAGE. 
 
 149 
 
 lay a power equivalent to 10 lbs., then in order that the weight 
 might be raised through one inch it would be requisite for the 
 power to move through a space of 50 inches in the same time. 
 This diminution of velocity in the inverse proportion of the 
 increase of force is the foundation of the common phrase, ap- 
 plied to machines, that what is gained in jpower is lost in 
 velocity; and we may regard it as another form of stating the 
 principle of virtual velocities in this particular case, or the 
 same as asserting that no efficiency is gained hy the intervention 
 of a machine. 
 
 133. Before closing this chapter we will briefly allude 
 to Ihe principle of the differential axle and Hunter s Screw. 
 On the wheel and axle we have seen that (Art. 100) 
 
 P _ radius of axle 
 W radius of wheel ' 
 
 or W=F 
 
 radius of wheel _ 
 radius of axle ' 
 
 from which it would appear that by suflSciently diminishing 
 the radius of the axle a given power P might be made to raise 
 a weight W of any magnitude we please. Practically how- 
 ever there is a limit to the thickness of the axle ; for if it be 
 made too small, the material of which it is made will not bear 
 the strain upon it, and it will break. 
 
 This imperfection isobviated in the 
 differential axle, the mode of action of 
 which will be sufficiently clear from 
 the figure, — the string from which W 
 is suspended by a pully passing round 
 the two axles B, in opposite direc- 
 tions : if The the tension of this string, 
 a, h, the radii of A, B, C respec- 
 tively, we shall have for equilibrium 
 2T=W, 
 
150 
 
 or THE MECHANICAL POWERS. 
 
 Pa+Tc=Tb; 
 
 a a 2 ' 
 
 W 2a 
 and the mechanical advantage =-^ = ^ , 
 
 which may be increased to any extent by making the axles 
 JB and (7 as nearly equal as we please without unduly re- 
 ducing the strength of the axle. 
 
 134. Again, in the case of the screw, it is obvious from 
 the expression 
 
 TF _ circumference of circle described by P .. . ,,g\ 
 P distance of two threads ^ " '' 
 
 that by diminishing the distance between two threads 
 sufficiently, we might obtain any mechanical advantaffs we 
 please ; but the distance between the threads must not be less 
 than .the thickness of the threads, .otherwise the companion 
 screws could not work together ; and further, if the thickness 
 of the threads be unduly diminished, they will not be able to 
 bear the strain upon them. This 
 difficulty is obviated in Hunter's 
 Screw, in which a screw A works 
 within another screwP; thus if c be 
 the radius of the circle described 
 by P, — b, a the distance between 
 two threads in the screws P, -4 re- 
 spectively, — then whilst P makes 
 one revolution, W will descend 
 through h, in consequence of the 
 screw B descending through i, but it will also rise through 
 
hunter's screw. 151 
 
 a in consequence of the screw A making one turn within 
 B\ i.e. TF will descend through i — a. 
 
 Whence P. P^s displacement = W. Wa displacement gives 
 
 P.2'irc=W.{h-a), 
 
 , W 2irc 
 ^""^-P'b^a' 
 
 and by making b as nearly equal to a as we please, the me- 
 chanical advantage may be increased to any extent without 
 imduly weakening the threads of the screws. 
 
DYNAMICS. 
 
 CHAPTER I. 
 
 INTEODUCTION. 
 
 1. A MATEEIAL particle has been defined to be a portion 
 of matter indefinitely small in all its dimensions. It has there- 
 fore no determinate form or volume, — but it has mass, it may 
 be subject to the action of force, and may exert pressure on 
 other particles. This conception of a particle is of course 
 conventional, — a result of arbitrary definition, — so that calcu- 
 lations respecting such a body cannot be at once practically 
 applied, since no bodies of which we have any experience 
 correspond to this idea. But a particle having no parts, its 
 motion is one and indivisible, and is therefore of a simpler 
 kind than that of a body of finite size, different points of 
 which might move differently. Hence we are led to con- 
 sider the motion of a particle preparatory to that of bodies of 
 finite size, and which have a real existence. The motion of 
 such bodies can be reduced to a dependence on that of par- 
 ticles, by the application of suitable principles, but in the 
 present treatise we do not propose to consider the motion of 
 anything but particles or bodies regarded as particles; for 
 example, a ball or a body of any kind, whenever it may occur 
 in the following pages, will be considered, so far as its motion 
 is concerned, as a particle coincident in position with the 
 centre of gravity of the ball or body. 
 
MEASURE OF VELOCITY. 153 
 
 2. When the position of a particle relative to certain 
 fixed points is being altered, it is said to be in motion, — and 
 its path is the line (straight or curved) along which it moves. 
 The tangent to the path at any point is the direction of the 
 particle's motion at that point. 
 
 Bef. Velocity is the term employed to express the degree 
 of swiftness or speed with which a body is moving, and this 
 velocity is said to be uniform when equal lengths of path are 
 passed over in equal intervals of time, however large or small 
 the intervals be taken; when this is not the case, the velocity 
 is variable. 
 
 When the velocity of a body is uniform, it is measured 
 by the space passed over in a unit of time,-^and is the same at 
 every instant whilst the motion continues uniform. 
 
 When the velocity is not uniform, it is measured at any 
 instant by the space which the body would describe in a 
 unit of time, if the body retained during that unit the same 
 velocity which it has at the instant when it is under con- 
 sideration. 
 
 3. An illustration may serve to make this mode of mea- 
 suring variable velocity clearer. The speed of a railway-train 
 is in general continually varying, and considering its motion at 
 any instant we should say that it was travelling at so many 
 (say 30) miles an hour, without much risk of being misunder- 
 stood; we should mean, not that it had travelled 30 miles 
 during the last hour, nor that it would travel 30 miles during 
 the next hour, hut that if it were to travel for an hour with 
 the speed it possesses at the instant considered, it would pass 
 over exactly 30 miles. 
 
 In fact, the velocity of a body at any given instant must 
 be regarded as a quality which it then possesses without any 
 
154 INTRODUCTION. 
 
 reference to its anterior or subsequent state, and without any 
 reference to causes whicli may have produced or may alter 
 the velocity. 
 
 This velocity may and will be influenced ^ow time to time 
 by different agents, but the velocity at. any instant we regard 
 as a quality which the body then possesses in the same sense 
 that it possesses mass and position. 
 
 When then we represent the velocity of a body by a 
 symbol, as v„ we mean (certain units of time and of length 
 being understood) that if the velocity continued of the same 
 intensity for a unit of time, the body would in that unit of 
 time pass over v units of length. 
 
 And the magnitude of this numerical representative of the 
 velocity («) will depend upon the magnitudes of the units of 
 time and space, and will vary with them, vis. directly as the 
 unit of time, and inversely as the unit of space. 
 
 Thus a velocity of 360 feet per minute is equivalent to 
 120 yards per minute, or to 6 feet per second, or to 2 yards 
 per second. 
 
 4. Formula Jor uniform motion. 
 
 Let s be the space described in time t by a body moving 
 with uniform velocity v, then since in each successive unit of 
 time the body passes over v units of length, we shall have 
 vt for the whole space passed over in t units of time, i. e. 
 s = v. t, — ^^a formula which holds true whether t be integral or 
 fractional. 
 
 5. We proceed next to explain how change of velocity 
 is measuiEed. 
 
 When the velocity of a body is continually increasing in 
 such a way that it receives equal increments of velocity in 
 equal successive intervals of time,, however large or small 
 
MEASURE OP ACCELERATION. 155 
 
 these intervals Tooay be, the body is said to be uniformly 
 accelerated, or the ucoeleration is said to be uniform. 
 
 When the velocity changes in any other way, the accele- 
 ration is variable. 
 
 Measure of acceleration. 
 
 When the acceleration is tiniform, it is measured by the 
 quantity by which the velocity is increased in a unit of time, 
 and is the same at all times during the motion. When the 
 acceleration is variable, it is measured at any instant by wJiat 
 would he the increase of velocity in a unit of time, supposing 
 the rate of increase of velocity to be uniform for that unit, 
 and of the same intensity as at the instant considered. 
 
 6. When then we express the acceleration of a body by 
 a symbol yj we mean (certain units of time and space being 
 understood) that if the rate of increase of velocity continued 
 of the same intensity for a unit of time, the velocity would be 
 increased by /"at the end of that unit. 
 
 A second is frequently taken as the unit of time, and a 
 foot of length, but as before in Art. (3), any other units m^ght 
 be chosen instead, and the numerical value of / for given 
 units of time and length being given, its numerical value for 
 any other assigned units of time and length may be found. 
 
 Thus, let f feet be the velocity generated in one second, 
 the acceleration being uniform, then €0/ will be the velocity 
 generated in 60 seconds, i. e. in one minute. 
 
 This means that the body at the end of one minute would 
 have acquired a velocity of 60 /per second. 
 
 Remembering then that when we use a minute as the 
 unit of time we must measure velocities by the spaces which 
 would be described in one minute, the velocity acquired would 
 
156 INTRODUCTION. 
 
 be 60 . 60 ./ per minute. Hence / feet being the measure of 
 the acceleration when one second is the unit of time, 60^./ 
 will be the measure of acceleration when a minute is the unit 
 of time. 
 
 Thus if the unit of time be altered, the numerical value of 
 the acGeleration will vary as the square of the unit of time, 
 and besides — as in the case of velocity, Art. (3) — if the unit of 
 space be altered, the numerical value oi f will vary inversely 
 as the unit of length. 
 
 Ohs. Retardation may in all cases be regarded as a nega- 
 tive acceleration. 
 
 7. The term force has been applied to any cause which 
 tends to move a body or to alter the state of its existing 
 motion. This conception of it renders it unnecessary to con- 
 sider the manner in which force is produced, whether it be 
 by the agency of living bodies, or the pressure of inanimate 
 substances, or by the intrinsic attraction of matter. We shall 
 regard force simply with reference to its effects, viz. the pro- 
 duction of motion in material bodies ; and this points directly 
 to the two particulars to which the student is requested to 
 give his attention in estimating the effects of force, — the matter 
 or mass moved, and the velocity and change of velocity pro- 
 duced. 
 
 8. We here introduce a new quality, viz. that of mass, 
 which is perhaps not familiar to the student. But experience 
 teaches us that equal efforts are not required to produce the 
 same motion in different bodies. It will probably be admitted 
 without hesitation that equal volumes of the same substance 
 would acquire equal velocities by the application to them of 
 equal forces for the same time ; but this would not be the 
 
MASS. 157 
 
 case witli equal volumes of different substances. In fact it 
 will frequently happen that when equal forces are applied 
 for the same time to bodies of diiferent substance and of 
 unequal volume, the velocity acquired by the body of greater 
 volume will be greater than that of the other, arid vice versd ; 
 so that the consideration of volume is not sufficient for the 
 comparison of bodies under this aspect, and it, is necessary to 
 introduce a new idea, vis. that of mass or massiveness, and 
 this must be regarded as a quality of matter sui generis, as 
 much so as its weight, form, volume, &c. 
 
 9. We give no definition of this new species of quantity, 
 which is a fundamental one in Dynamical science ; for such 
 definitions as might be given would be as illusory as those 
 which might be given of time, space, and many other species 
 of magnitudes. 
 
 But it is necessary clearly to define equality between quan^ 
 titles of this new species, so that in estimating the mass of a 
 body it may be referred to the like quality of some other body 
 taken as a standard. 
 
 "We give then the following definition of eqirnl masses : 
 
 Bef. " The masses of two particles are said to be equal 
 when two equal forces acting on them similarly for the same 
 time generate in them equal velocities." 
 
 The notion of equality of mass of two "bodies will readily 
 lead to that of bodies whose masses have any assigned ratio. 
 Thus if the masses of two bodies A and B are said to be in 
 the ratio m : m, it is meant that A and B might be divided, the 
 former into m equal parts and the latter into n equal parts, 
 any two of which parts are of equal mass and satisfy the 
 above definition. 
 
158 INTEODUCTION. 
 
 Dbf. The mass of a unit of volume of any substance is 
 called its density, — so that if m be the mass of a body whose 
 volume is V and density p, we have m = Vp. 
 
 10. If the notion of mass is not familiar to the student, 
 he will; perhaps consider the account of it given above in Arts. 
 (8, 9) vague and unsatisfactory. The same vagueness attaches 
 to any species of quantity or quality till the conception; of it 
 is impressed on the mind by continued experience, and this 
 holds more especially with such qualities as are not obvious 
 to the senses. For example, form and volume being qualities 
 obvious to the eye, the conception of form and volume is 
 much more readily acquired than that of hardness, which 
 requires farther experience to familiarize the conception of it. 
 So the conception of the mass or massiveness of matter, not 
 being obvious to the sight or touch requires further experience 
 before it becomes familiar to the mind. 
 
 We may regard the massiveness of matter as that quality 
 which enables it to act upon other matter isolated from 
 itself: for instance, in the case of bodies at the earth's sur- 
 face it is that quality which subjects them to the influence 
 of the earth's attraction and causes in them the quality which 
 we call weight. 
 
 11. The unit of mass may be assumed at our convenience; 
 thus we might take the ma^s of a cubic inch of lead fori our 
 unit if it be convenient to do so under any particular circum- 
 stances. 
 
 And when we express the mass of a body by a symbol m 
 we mean that the body has m times the mass of that body 
 whose mass we have taken for our unit of reference. See 
 Art. (44). 
 
ACCELEEATING AND MOVINa POECE. 159 
 
 12. Momentum. 
 
 Def. If m and ® be the numerical measures of the mass 
 and velocity of a body, the product mv is called the momentum 
 of the body. 
 
 The momentum of a particle must be viewed as a quality 
 sai generis, and is to be compared only with the same quality 
 of other particles, and apart from any external agency which 
 may have been instrumental in producing it. 
 
 Ohs. The velocity considered in Arts. (2, 3) is sometimes 
 called absolute velocity, having been defined and measured 
 with reference to points fixed in space: and this is distin- 
 guished from r dative velocity, which is the. term applied to the 
 same quality defined and measured with reference to points 
 which maintain an invariable position with regard to- one 
 another, but which are not necessarily fixed in space. 
 
 A similar remark applies ta ahsolute and relative accele- 
 ration, and to ahsolute and relative momentum. 
 
 13. Supposing then a particle's geometrical and dynamical 
 state to be defined at any instant by a knowledge of its mass, 
 position, velocity, acceleration and direction of motion, we 
 proceed tO' examine and measure the forces to which it, is 
 subjected. 
 
 It is often' convenient to consider the transfer of a body 
 from one position to another without introducing any con- 
 sideration of the mass of the body, i.e. to treat of the velocity 
 and acceleration exclusively of the mass. 
 
 Bef. When we regard a force undet this aspect, with 
 reference that is to the acceleration it can produce, or its 
 power of accelerating a given body, we speak of it as an 
 " accelerating; force," and. we measure the accelerating force 
 
160 INTRODUCTION. 
 
 simply hj the acceleration of the body, i. e. hy the velocity 
 which it can generate in the body in a unit of time, if 
 nniform; or if variable, by the velocity which it would gene- 
 rate in a unit of time, if it acted for a unit of time with the 
 same intensity as at the instant considered. 
 
 14. Ohs. The term acceleration or power of acceleration. 
 would better express that particular effect of a force which is 
 here considered, but the term accelerating force has been long 
 sanctioned by usage, and the student is here cautioned that 
 the term is used in the above sense. For example, the phrase 
 "A body subject to an accelerating force f, &c." must be un- 
 derstood to mean " A body subject to a force which can pro- 
 duce an acceleration / in that body, &c." 
 
 Thus when we say the accelerating force on a body is f, 
 or a body is subject to an accelerating force^ we mean that 
 / is the acceleration of the body at the instant considered, 
 or, that velocity is being generated in the body at that instant 
 at the rate of/ units of space per unit of time. 
 
 15. Def. A force considered as communicating motion to 
 matter, i.e. regarding both the amount of mass moved and 
 the amount of motion which it can communicate to it, is called 
 amoving force; and it is measured by the momentum which 
 it can generate in a unit of time, the intensity of the force 
 remaining constant for that time. 
 
 Thus, if F be the symbol which represents a moving force, 
 and f represents the accelerating force of jF on a mass, m, i. e. 
 a f represent the velocity which F can generate in m by 
 acting uniformly upon it for a unit of time, we have F=mf. 
 
 Hence we see that a moving force is expressed by the 
 product of the number of units of mass in the body, and the 
 
IMPULSIVE FORCE. 161 
 
 number of units of acceleration wliich it can produce in the 
 body. 
 
 Ols. This is sometimes given as the definition of moving 
 force, i.e. moving force has sometimes been defined to be 
 the product of the mass into the accelerating force. 
 
 16. The definitions above given of mass, momentum, 
 accelerating force, moving force, must be looked upon as 
 arbitrary ; but we accept them as convenient terms to employ 
 in expressing the laws of the action of force upon matter and 
 in deducing from them their legitimate consequences. For a 
 knowledge of these laws we must have recourse to experiment, 
 for we can have no d, priori knowledge of the constitution of 
 matter, or of the principles which regulate and modify its 
 dynamical state or condition, 
 
 17. Impulsive Force. 
 
 The efiect of a force to which our attention is to be 
 directed is the motion produced in a given mass; and this 
 effect we regard as produced gradually in all cases. In other 
 words, we consider that some time is necessary during which 
 a force produces its effect. This time may be finite and 
 appreciable, as when a body is pulled along a plane, or when 
 a body falls to the earth under the action of gravity, or when 
 a railway-train gets up its speed from a state of rest by the 
 action of steam-pressure. But cases frequently occur in which 
 the time required for the effect of a force to manifest itself 
 is very small and, so to speak, inappreciable; as for example, 
 when a body is put in motion by a blow, almost instanta- 
 neously. 
 
 When a force requires a finite and appreciahle time in 
 order to produce an appreciable motion, it is not unfrequently 
 p. M. 11 
 
162 INTEODUCTION. 
 
 called a finite force, as in the cases first mentioned. But 
 wlien motion is produced by a blow or impulse, as in the 
 latter case, in an indefinitely small time, the force is gene- 
 rally called impulsive; but still we regard the effect as pro- 
 duced by the action of a force operating for a very short 
 
 time. 
 
 In such cases the impulsive force, or the force of the 
 blow, is measured by the momentum generated in the body by 
 the impulse. 
 
 18. To illustrate this, we may suppose a ball B at rest to 
 be suddenly put in motion by a ball A striking it. The balls 
 will be in contact for a short time (t suppose), and during this 
 interval A will press B with a force varying in intensity from 
 the beginning to the end of the interval t. The moving force 
 which acts on B will thus be varying, but we may practically 
 consider it to have a mean intensity and to remain uniform 
 during the time of contact. If we call this moving force F, 
 and /the acceleration which F can produce in the ball B (the 
 mass of which we will call m), and v the velocity acquired by 
 B at the end of the time t, we shall have, Arts. (6, 15), 
 
 V =fr, and F= mf 
 consequently Fr = mfr = mv. 
 
 Now V the velocity acquired by B is finite, and therefore 
 the momentum mv is finite, so that although t is in general 
 so small as to be inappreciable, yet F is so large as to render 
 the product Ft finite, and, we take this finite product to he 
 the measure of the impulsive force, — in other words, if P be 
 an impulsive force which produces a velocity « in a mass m, 
 then F= mv. 
 
 19. Before stating the laws of motion, we proceed to 
 give some explanation of the geometrical representation of the 
 
EEPEESENTATION OF VELOCITY. 
 
 163 
 
 position and motion of a particle; and for the sake of simpli- 
 city we -will in the following Articles (19 — 24) suppose the 
 motion to take place in one plane (that of the paper suppose). 
 
 If two lines Ox, Oy be drawn in 
 the plane of motion inclined at any 
 given angle, the position of any point 
 P may be simply defined with re- 
 ference to these lines (as fixed lines 
 of reference), by drawing two lines 
 through P, one of them PN parallel 
 
 to Ox, the other PM parallel to Oy : if the distances OM, 
 ON corresponding to any point are known, the position of 
 the point is easily determined; for we have simply to draw 
 lines through the points M, N parallel respectively to Oy 
 and Ox, and the point in which these lines intersect is the 
 geometrical position of P. 
 
 The point M is called the projection of the point P on the 
 line Ox, or the position ofP referred to the line Ox, and simi- 
 larly ^is \h& projection of P on the line Oy, or the position of 
 P referred to the line Oy. 
 
 Note. The above mode of representing the position of a 
 point will be familiar to the student who is acquainted with 
 co-ordinate geometry as the method of co-ordinates, OM and 
 0^ being the co-ordinates of P measured along Ox and Oy. 
 
 20. Resolution and composition of velocities. 
 
 Further, if a particle be moving 
 with uniform velocity in a given 
 direction (as PQR), so that P, Q, 
 R are the positions of the particle at 
 given instants, and if we regard those 
 points P, Q, R as determined by 
 
 11—2 
 
164 INTRODUCTION. 
 
 their projections on two fixed lines Ox, Oy, then will p, q, r he 
 the projections of P, Q, R on Ox, and we readily see hy simple 
 geometry that the ratio of pq : pr is the same as that of 
 PQ:PB, whatever he the intervals of time in which the lengths 
 PQ, PR are described by the particle. That is, if the particle 
 move uniformly, its prelection on a given line Ox will also 
 move uniformly, — not with the same velocity as the particle, 
 hut with a velocity which bears to that of the particle a ratio 
 dependent only on the inclination of the direction of motion 
 of the particle to the two lines of reference Ox, Oy. The 
 same is obviously true of the projection of the particle upon 
 the other line of reference Oy. 
 
 The velocity of the projection of P along Ox is called the 
 velocity of P resolved along Ox, or the velocity of P referred to 
 Ox ; and similarly with respect to the velocity referred to Oy. 
 
 21. Now since we regard the velocity of a particle at 
 a given instant as a quality which the particle then possesses, 
 without any reference to the time during which it retains that 
 velocity, or the space through which it moves in consequence 
 of it, and also without reference to any causes which may 
 subsequently modify it, — we may represent 
 the velocity of a particle P by a line PQ 
 drawn in the direction of motion, and pro- 
 portional to the velocity in magnitude; 
 and if a parallelogram be constructed, of which PQ is the 
 diagonal, and the sides of which (viz. PR, PS) are in the 
 direction of known lines of reference, the sides PR, PS will 
 represent the velocity of P resolved along those lines of refer- 
 ence severally. ^ 
 
 Theorem. In other words, we have this theorem : " If a 
 straight line PQ which represents the velocity of a particle be 
 
GEOMETEICAL THEOREMS. 165 
 
 made the diagonal of a parallelogram PQB8, whose adjacent 
 sides PR, PS are in assigned directions, the resolved velocities 
 in the direction of the sides will he represented by those 
 sides respectively." 
 
 And conversely, 
 
 " If the resolved velocities of a particle in two given direc- 
 tions be represented by two lines PB, P8 drawn from a point 
 P, the actual velocity of the particle will be represented in 
 magnitude and direction by the diagonal of the parallelogram 
 constructed on those two lines as adjacent sides." 
 
 22. In the preceding Article we have represented velo- 
 cities as to magnitude and direction by lines, and in a similar 
 manner we may represent accelerations by lines, — and we may 
 regard an acceleration as resolved' in given directions in the 
 same way as we supposed a velocity to be resolved in given 
 directions; and with this understanding we shall have the 
 following theorems respecting acceleration analogous to the 
 preceding ones respecting velocity, viz. 
 
 Theorem. " If a line AD, which 
 represents the acceleration of a 
 particle, be made the diagonal of a 
 parallelogram ACDB, whose ad- 
 jacent sides A 0, AB are in assigned 
 directions, the resolved accelera- 
 tions in the directions of the sides will be represented by 
 those sides respectively." 
 
 And conversely, 
 
 " If the resolved accelerations of a particle in two given 
 directions be represented by two lines A C, AB drawn from a 
 
166 INTEODUCTION. 
 
 point A, the actual acceleration of the particle will be repre- 
 sented in magnitude and direction by the diagonal of the 
 parallelogram constructed on those two lines as adjacent 
 sides." 
 
 23. The theorems of the two preceding Articles may- 
 be called the Parallelogram of Velocities, and the Parallelo- 
 gram of Accelerations. They are analogous to the Paral- 
 lelogram of Forces in Statics, and admit of the same exten- 
 sion as the latter theorem, so far as composition and resolution 
 are concerned ; but the student must bear in mind that these 
 two theorems respecting velocity and acceleration form part 
 of a purely conventional mode of representing geometrically the 
 position and motion of a particle ; and he must be careful not 
 to confound the meaning of lines which in one problem may 
 be employed to represent velocity, with the meaning of other 
 lines, which in the same or other problems may be taken to 
 represent acceleration, and vice versd. 
 
 24. The results of the parallelo- 
 gram of velocities may be stated as fol- 
 lows, with algebraic symbols — ^results 
 which may easily be obtained by trigo- 
 nometry. A velocity V, in a direction 
 inclined at angles a, yS to the lines Ox, 
 Oy is equivalent to the velocities 
 
 F .^i"^ , andF ''" " 
 
 sm (a + fi) sin (a + ^) ' 
 
 resolved in the direction of Ox and Oy respectively. 
 
 And conversely, if these component velocities in direction 
 of Ox, Oy be represented by X and Y, the actual velocity V 
 and its direction will be determined from the equations 
 
GEOMKTEICAL THEOREMS. 167 
 
 X^-V ^'"^ Y=V—^^^"' 
 
 sin (a + ;8j ' sin (a + /S) ' 
 
 which give V' = X'+Y' + 2XYcos {a + ^)\ 
 
 X+rcos(a + jS) J 
 
 which two equations determine the magnitude and direction of 
 the actual velocity, — the angle (a + ^) between the fixed lines 
 Ox, Oy being a known angle. 
 
 The same formulae, substituting acceleration for velocity, 
 will hold good for the resolution and composition of accelera- 
 tion. 
 
 25. The advantage of the mode of geometrical repre- 
 sentation explained in Articles (1 9 — 24) will become obvious 
 to the student when he has become acquainted with the laws 
 of motion, and the application of them to determine the 
 position and motion of a particle when acted on by known 
 forces. 
 
 Obs. We have supposed, as was before mentioned (19), 
 that the motion is entirely in one plane; when this is not the 
 case, the method must be extended by taking three lines of 
 reference in space analogous to the method of co-ordinates in 
 geometry of three dimensions; but in the present treatise we 
 shall have occasion to consider but few cases of motion which 
 may not be regarded as taking place in one plane. 
 
 26. Having explained the mode of representing the 
 motion of a particle geometrically, we proceed to enunciate 
 and illustrate certain principles deduced from observation and 
 experiment which are commonly called laws of motion, and 
 according to which the motion of a body considered as a 
 particle are calculated. 
 
168 INTEODUCTION. 
 
 FIRST LAW OF MOTION. 
 
 27. A particle if at rest will continue at rest, and if in 
 motion will move in a straight line with uniform velocity unless 
 it is acted on hy an extraneous force. 
 
 This law is sometimes referred to as the law of Inertia or 
 the principle of Inertia. It expresses the fact that a particle 
 of matter has no power within itself of altering or influencing 
 its own state of rest or motion. 
 
 Of this principle no direct proof can be given, but it may- 
 be rendered probable by such experiments as the following. 
 If a ball be projected along a smooth pavement it will con- 
 tinue in motion for a considerable time, and its path will be 
 more nearly a straight line the smoother the pavement is; 
 but the friction will gradually reduce it to rest : if it be pro- 
 jected along a sheet of ice, it will continue longer in motion, 
 and will move more uniformly. Such experiments may suggest 
 the inference that if all extraneous force could be removed, 
 the ball would go on for ever with uniform velocity. 
 
 28. Having established the principle that a particle can- 
 not put itself in motion, nor alter in any manner the nature 
 of its own motion when it is in motion, we next require some 
 principle which will enable us to calculate the effects of 
 forces on a particle in motion. The experiments and re- 
 searches of philosophers have led to the following, which 
 may be called 
 
 THE SECOND LAW OF MOTION. 
 
 29. When a particle is in motion under the action of any 
 force, the acceleration of the particle estimated in any assigned 
 
SECOND LAW OF MOTION. 169 
 
 direction is wholly due to the force resolved in that direction, — 
 and is the same in intensity as if that resolved force alone acted 
 on the particle at rest. 
 
 Thus for example, if the particle F be moving with any 
 velocity v in the direction FV, and y/ 
 if X, Yhe the forces acting upon 
 F at that instant resolved in direc- 
 tions parallel to two fixed lines Ox, 
 Oy, — this second law asserts that 
 (at the instant under consideration) 
 the accelerations of F estimated in 
 
 directions parallel to Ox and Oy are the same as would arise 
 from the separate and independent action of X, Y upon F at 
 rest, in direction of Ox and Oy. 
 
 By the acceleration due to a force is meant of course the 
 acceleration which that force is capable of producing in the 
 particle. And if several forces act simultaneously on the par- 
 ticle, the force mentioned in the enunciation of the second law 
 must be taken to mean the resultant of the several forces 
 which act upon the particle: this resultant being determined 
 in the same way as the resultant of statical forces is de- 
 termined. 
 
 The enunciation of this second law further implies the fact, 
 that the accelerating power of a given force upon a particle (or 
 what we have before called the accelerating force) estimated 
 in the direction of its action, is of the same intensity what- 
 ever be the dynamical state of the particle, i. e. whatever be 
 the velocity and direction of the particle's motion, — or, the 
 same as it would.be if the particle were for an instant at rest. 
 
 30. The principle stated in the second law of motion will 
 be sufficient (theoretically speaking) when the accelerating 
 
170 INTRODUCTION. 
 
 forces acting upon a particle are given, to enable us to deter- 
 mine the motion of the particle, i. e. to determine its position and 
 velocity at any time ; for, we should only have to calculate its 
 position and velocity referred to two fixed lines as Ox and Oy 
 (Art. 19, &c.) (or referred to three lines fixed in space if the 
 motion be not in one plane), and when its motion referred to 
 these lines is known, its actual position and velocity are 
 known. 
 
 It will however frequently be the case in nature that the 
 forces on a particle will vary with the position of the particle, 
 and thus its motion will indirectly affect the forces which act 
 upon it. To determine the motion of a particle generally will 
 require the processes of the Integral Calculus, but in the pre- 
 sent treatise we do not propose to consider any motions which 
 require for 'their calculation anything beyond ordinary alge- 
 bra; such for example as arise from the action of uniform 
 forces or from impulsive action, or these combined. 
 
 31. Illustrations of the second law of motion. 
 
 Experiments such as the following may be mentioned as 
 illustrating and confirming the second law of motion. 
 
 If a ship be moving uniformly, a ball when thrown with 
 the same force will go to equal distances from the ship, 
 whether it be thrown towards the bow or the stern, or at 
 right angles to the direction in which the ship is moving, A 
 ball let drop from the top of the mast will strike the deck at 
 the foot of the mast, and will fall in the same time, whether 
 the ship be at anchor or moving uniformly. A ball let drop 
 from the top of a railway-carriage in uniform motion, will 
 strike the floor of" the carriage at the point directly beneath 
 the point from which it started. A pendulum will vibrate in 
 
SECOND LAW OP MOTION. 171 
 
 the same time from east to west, as from north to south, or 
 in any other direction ; thus shewing that whilst it is carried 
 uniformly in one definite direction by the earth's rotation, its 
 motion relatively to bodies on the earth's surface which have 
 the same motion as the pendulum arising from the earth's 
 rotation, is uninfluenced by the motion thence arising. 
 
 32. Experiments such as liese, of course, do not ^rove 
 the law. Strictly speaking it could only be proved by shew- 
 ing it to be true for every individual case that can occur, 
 which is manifestly impracticable. But when the results of 
 numerous and intricate calculations based upon it are invari- 
 ably found to agree with observation, we arrive at a moral 
 conviction of its truth. And the principle itself having been 
 obtained by induction from a considerable number of facts and 
 observations, we employ it with confidence in deducing other 
 consec[uences from it. 
 
 33. The mode of obtaining the magnitude of the accele- 
 ration which a given force is capable of producing in a given 
 particle will be explained presently (see third law of motion); 
 and assuming the principle which has been stated in the second 
 law of motion, and illustrated in the preceding articles, we 
 shall know how to obtain the magnitude of the acceleration 
 of the particle, estimated in any proposed direction, when we 
 know the forces which are acting upon it. Now the effect of 
 an acceleration on a particle is to create or modify velocity 
 in the particle, and we may regard the velocity with which 
 a particle is at any instant animated, as the accumulation 
 of the effects of the accelerating forces which have acted 
 upon it during the successive portions of the interval of time 
 during which it has been in motion. This consideration will 
 
172 INTRODUCTION. 
 
 readily lead us to the following conclusion as a necessary- 
 consequence of the second law of motion stated in Article 29, 
 viz. the velocity of a particle estimated in a given direction 
 is wholly due to the acceleration which has operated on it in 
 that direction. 
 
 34. In order to ohtain actually the velocity acquired by 
 a particle by the action of given forces, we shall in general 
 require (as before remarked) the integral calculus: but we 
 will here give the solution of a particular problem of frequent 
 occurrence, which can be readily inferred from the second law 
 of motion. 
 
 35. If there he simultaneously impressed on a particle 
 two velocities which would separately lie represented hy the 
 lines AB, A G, the actual velocity will he represented hy the line 
 AD, which is the diagonal of the paral- 
 lelogram of which AB, A G are adja- 
 cent sides. 
 
 Let the motion of the particle be 
 referred to the directions AB, AG, 
 then we may suppose the particle at 
 rest at A to receive simultaneously 
 two blows in directions AB, A G respectively, which would (if 
 they operated independently of each other) generate velocities 
 represented by AB,AG. Complete the parallelogram ABDO. 
 Then since the instant after the blows are communicated to 
 the particle it is subject to no force, it must move with 
 uniform velocity in some straight line (by first law of motion), 
 and this straight line must be such that the velocity along it 
 when referred to the lines AB, A G will be represented by the 
 lines AB, AG (by the second law). The resultant velocity 
 
PAEALLELOGEAM OP VELOCITIES. 173 
 
 then must be represented by AB the diagonal of the paral- 
 lelogram ABDG. 
 
 36. In the problem of the previous article instead of 
 supposing two blows to be given to the particle simultane- 
 ously, we may suppose one blow given to the particle already 
 in motion ; for example, if the particle be moving in direc- 
 tion AC with a velocity represented by AG, and at the 
 instant the particle is at A let a blow be given to it in 
 direction AB, capable of producing on the particle at rest a 
 velocity represented by AB: the actual velocity as regards 
 direction and magnitude (by the same reasoning as the above) 
 will be represented by AD. 
 
 Hence it appears that velocities (regarded as the effects 
 of impulses) may be compounded in the same way that 
 statical forces are compounded by the polygon of forces, and 
 the same theorem mutatis mutandis will hold good for velo- 
 cities as for statical forces. 
 
 37. The theorem stated and proved in Article (35), may 
 for convenience be referred to as the dynamical parallelogram 
 of velocities ; — it is a consequence of the second law of motion, 
 and exhibits the application of that law to find the effect of 
 one or more impulsive forces — those effects being expressed 
 in accordance with the geometrical mode of defining and 
 estimating motion previously explained (Art. 19 — 25). The 
 student will be careful to distinguish it from the geometrical 
 parallelogram of velocities explained in Art. (20), which is 
 simply a geometrical convention. 
 
 In a similar manner from observing the analogy between 
 the statical parallelogram of force and the second law of 
 motion, we may regard the second law as the dynamical 
 
174 
 
 INTRODUCTION. 
 
 parallelogram of accelerations; carefully distinguishing it 
 from the geometrical parallelogram of accelerations stated in 
 Art. (22). 
 
 38. We may, if we please, regard any velocity with which 
 a particle is animated as remaining permanently impressed 
 upon it, and when force acts on the body, the velocity arising 
 from the action of that force may be regarded as superadded 
 (so to speak) to the existing velocity — so that the actual 
 velocity of the particle at any time in a given direction will 
 he the algebraic sum of the velocities which have been im- 
 pressed upon it in that direction, 
 
 39. The following experimental illustration may be given 
 of the dynamical parallelogram of velocities. 
 
 Let AB represent the deck of a ship which is moving 
 uniformly parallel to itself from left to right, and let a body 
 on the deck have a velocity communicated to it, which if the 
 ship were at rest would make it move uniformly from C to B 
 along the line CD in the same time that the ship moves from 
 AB to the position AB'. 
 
 It is found hy experience that the body moves on the deck 
 relatively to it in exactly the same way as if the ship were at 
 rest, i.e. (drawing the parallelograms as in the figure) if in any 
 time T the line GB would have been brought to Gfl, by the 
 motion of the ship, and GP be the space which the body 
 would have moved through in the same time t if the ship had 
 
PAEALLELOGEAM OF VELOCITIES. 175 
 
 been at rest, then it is found that P, is the position of the 
 body {CiP = GP) at the end of the time t, whatever be the 
 magnitude of t. 
 
 Thus, in reality, the velocity of the ship has been com- 
 pounded with that of the body, and the body has described 
 in space the line CD' uniformly and in the same time that the 
 point on deck has moved to C". 
 
 This confirms the dynamical parallelogram of velocities, 
 and by inference the second law of motion also, so far as one 
 experiment can do so. 
 
 40. Thus we see that if a body be moving along with a 
 space which moves uniformly, and if any velocity be impressed 
 on the body, the motion of the body relatively to that space 
 will be the same as if the body and the space had been origi- 
 nally at rest; and more generally (if the second law of motion 
 be true) we infer that if several bodies be in uniform motion, 
 but be at rest relatively to each other; and if any force acts on 
 one of them, the motion of this one relatively to the others is 
 the same as if they had been all originally at rest.— Or we may 
 state this principle as follows: "If all the points of a system 
 have uniform and equal velocities and move in parallel direc- 
 tions, and if one of these points or particles be acted on by 
 any force, its motion relative to the other particles will be the 
 same as if the common motion of the system did not exist, 
 and the particle in question were acted on by the same force 
 acting in the same direction." 
 
 And from hence also it is obvious that we may (when we 
 find it convenient to do so) impress any the same uniform 
 velocity on each of the bodies composing a system, without 
 affecting the motion of these several bodies relative to each 
 other. 
 
176 TNTEODUCTION. 
 
 41. When the acceleration of a body for all successive 
 instants is known, the motion of the body can be calculated as 
 has been stated before. Now experience shews that the ac- 
 celerations produced In different bodies by equal forces are not 
 the same. We require then some principle based upon experi- 
 ment which will enable us to determine the acceleration of a 
 body of given mass when acted on by a given force or pressure. 
 The principle required for this purpose is called the third law 
 of motion, and may be stated thus. 
 
 THIRD LAW OF MOTION. 
 
 42. When a force or pressure acts on a particle, the moving 
 force on the particle is proportional to the force or pressure 
 acting upon it. 
 
 That is, if P, P' be two forces measured statically (viz. 
 
 by the weights -they would respectively support) acting on 
 
 two particles whose masses are m, m', and if f f be the 
 
 consequent accelerations of the two particles, then 
 
 P:P'::mf:mf; 
 
 or P °= mf 
 
 Since our units of force, mass, and acceleration are arbi- 
 trary, we may for convenience make P' =l,f'= 1, m' = 1, and 
 we shall then obtain P= mf. In other words, if we take our 
 unit of mass to be such that a unit of force acting upon it 
 would produce in it a unit of acceleration, then referred to 
 these units the number expressing the force will be equal to 
 the product of the numbers which express the mass and 
 acceleration. 
 
 43. If W be the weight of a body whose mass Is m, and g 
 the accelerating force of gravity at the surface of the earth 
 (I.e. the acceleration which the attraction of the earth, acting 
 
THIRD LAW OP MOTION. 177 
 
 freely on the body m vacuo, would produce in the body), we 
 shall have by the previous article W= mg = Vpg. (Art. 9.) 
 
 The numerical Value of g must of course be determined by 
 experiment, and the observations made upon pendulums are 
 those which give the most trustworthy results. They are 
 however of too refined a character to be introduced here ; and 
 it may be sufficient for the present purpose of the student to 
 state, that if a foot and a second be taken for the units of 
 space and time, the numerical value of g, in the latitude of 
 London, is 32"19 or 32*2 nearly. 
 
 The value of ^ is found by experiment to be slightly differ- 
 ent at different places on the earth's surface, but the variation 
 is so small, that we may for all ordinary purposes assume the 
 value of ^ to be that just given. 
 
 Or we may state the above result respecting the value of 
 g thus : 
 
 A body falling freely from rest in vacuo under the action of 
 gravity will, in one second from the beginning of its motion, 
 have acquired a velocity of 32 '2 feet per second. 
 
 44. If we call the moving force the dynamical measure of 
 a force, the third law of motion establishes a connexion 
 between the statical and dynamical measures of force, and 
 asserts that the statical measure is proportional to the 
 dynamical measure. 
 
 Considering the equations 
 
 m = Vp, W=mg=Vpg; 
 we see from the former that the unit of mass would be the mass 
 ofahodg of a unit of volume and a unit of density ; and from 
 the latter, since ^=32 '2 when a foot and a second are taken 
 for the imits of space and time respectively, the unit of weight 
 p. M. 12 
 
178 INTRODUCTION, 
 
 is the weight of a body of the unit of density and of volume 
 equal to the S^'^*'^ part of the unit of volume. 
 
 The density of distilled water is generally taken as the 
 unit of density, and a cubic foot as the unit of volume. 
 
 The weight of a cubic foot of distilled water is 1000 oz. 
 avoirdupois, nearly, 
 
 45. The equation P=mf must be always understood in 
 accordance with the explanation given in Article (42). As a 
 further illustration, we will apply it to the following problem. 
 
 A body weighing 24 lbs. is moved by a constant force, which 
 generates in a second a velocity of ^ feet per 1" ; find what weight 
 the force would statically support. 
 
 If we take m to represent the mass of the body and P for 
 the number of lbs. the force would support, g the accelerating 
 force of gravity, we have 
 
 24 = mg, 
 P=mf 
 
 m being the same in both equations. And by the question, 
 /"the acceleration produced by Pin the mass m is represented 
 by 3, a foot and a second being the units of space and time, — ■ 
 and with the same units, g the accelerating force of gravity 
 is = 32-2. 
 
 f 3 
 
 Whence P= *-24 =-——24 = 2-236lbs. nearly ; that is, the 
 g 32-2 •' ' 
 
 force which acts on the body would support in equilibrium 
 
 a weight equal to 2'236lbs. 
 
 46. Action and Heaction. 
 
 If two bodies A and B are in contact and at rest, we know 
 from statical principles that the pressure which A exerts upon 
 B is equal in magnitule, and opposite in direction to that 
 
ACTION AND EEAOTION. 179 
 
 which B exerts upon A ; and again, if two hodies A and B at 
 rest are connected by a fine thread, the strain which the 
 thread exerts upon one of the bodies is equal in magnitude 
 and opposite in direction to that which it exerts upon the 
 other. The question will arise, " Is this the case when the 
 bodies are in motion ? or, if the mutual pressures which they 
 exert on each other are not equal, what relation subsists 
 rbet ween them?" And to these questions (which must arise 
 in all problems where there is any mutual action between the 
 different parts of a system of bodies), the principles which 
 we have already stated afford no .satisfactory answer. It is 
 a^sifmed, however, that when one particle acts on another 
 particle, in motion as well as af rest, the second exerts on 
 the first a force equal in magnitude and opposite in direction 
 to that which the first exerts on the second. If the force 
 which the first exerts on the second be called " action" that 
 which the second exerts on the first may be called "reaction" 
 and the principle just stated may in other wo;rds be expressed 
 thus: "Whenever one body A acts on another B, the latter 
 reacts on the former, and this action and reaction are equal 
 in magnitude an^ opposite in direction." 
 
 47. The action here spoken of may be of any kind wh9,t- 
 ever ; as for example, when two bodies in motion or at rest 
 press against each other, their mutual pressures are equal and 
 opposite ; or in other words, the action and reaction are equal 
 in;magnitude and opposite in direction. Or again, when two 
 particles move in any manner connected by a string, the /orce 
 which the string exerts on one is equal a,nd opposite to the 
 -force it exerts on the other. Or .again, if two particles attract 
 or repel each other, the dynamical measure of the force which 
 one of them A exerts upon the other B, is. equal to that whicli 
 
 U—2 
 
180 INTRODUCTION. 
 
 B exerts upon A. This principle is frequently embodied In 
 the brief statement that " Action and Reaction are equal and 
 opposite." 
 
 48. For some illustration of the third law of motion, we 
 may refer to the observations made with Atwood's machine 
 (Arts. 80 — 82) ; but the motions of the heavenly bodies afford 
 the most interesting as well as the most searching test of the 
 truth of the dynamical principles which are employed in 
 investigating them. 
 
 It has been before remarked, that the laws of motion are 
 enunciated and asserted to be true only with respect to 
 particles, — and of course, as we have no practical experience of 
 particles, in the mathematical sense of the word, the student 
 must not expect to find them proved with that degree of 
 strictness which attaches to geometrical demonstration. He 
 is recommended for the present to accept them as conclusions 
 which have been arrived at'by philosophers after much painful 
 inquiry and observation, and not to trouble himself much 
 with the particular experiments which may be said to suggest 
 these laws, or with the calculations of more complex phe- 
 nomena which are based upon them, till he has grasped their 
 meaning, and applied them to a variety of problems. He will 
 then be able more fully to appreciate the bearing of particular 
 experiments on the principles which they are intended to 
 illustrate and confirm. But as no individual experiment will 
 involve one law of motion to the exclusion of the others, the 
 laws of motion must be taken as a whole, — and when we find 
 the observations of numerous and complex phenomena agree- 
 ing with calculations based upon these principles and involv- 
 ing them in every variety of combination, we arrive at a moral 
 conviction of their truth. 
 
( 181 ) 
 CHAPTER II. 
 
 OF UNIFORM MOTION AND COLLISION. 
 
 49. "When a body, regarded as a particle, is subject to 
 no extraneous force it moves with uniform velocity in a 
 straight line (first law of motion). If then v represent this 
 uniform velocity, and s be the length of path described or 
 passed over in any interval of time t, we shall have s = vt; 
 which is the formula for uniforni motion. 
 
 The equation s = vt which connects the three quantities 
 s, V, t will still be true if the path of the body be curvilinear, 
 provided the velocity be uniform : but when the path is not 
 a straight line there must be some force acting on the body 
 which deflects it from a rectilinear path ; and if the velocity 
 be uniform, this force must always act perpendicularly to the 
 direction of the body's motion at any time, and the magnitude 
 of this force will depend upon the curvature of the path. 
 This kind of motion however we do not propose to discuss. 
 
 50. The position, velocities, and direction of motion, of two 
 particles at any time being given, to find after what interval 
 they will he at an assigned distance 
 
 from each other, and to determine their 
 position at that time : the motion being 
 in one plane. 
 
 Let A, B be the position of the 
 particles at first, and AO, BO the 
 directions of their motion. Take A 
 to represent the velocity of A, and BT, on the same scale, 
 
182 OF tTNIPORM MOTION AND COLLISION. 
 
 to represent that of B. Complete the parallelogram A C anrl 
 join CT. Let a circle with centre and radius equal to 
 the proposed distance, cut CT in D: join OD and com- 
 plete the parallelogram PD. Then will P, Q be contemporary 
 positions of the particles originally at A, B, and the time of 
 moTing from Aio P: that from A to 0:: CD: CT. 
 
 For by the construction 
 
 PO: QT=QD: QT=BC : BT= AO : BT; 
 
 :. AO - PO : BT- QT= AO : BT; 
 
 le.AP:BQ = AO:BT; 
 
 that is, AP, BQ are in the proportion of the velocities of the 
 particles, and therefore they are simultaneously at P, Q, and 
 the distance PQ is equal to OD, the distance proposed ; and 
 further, time of moving from ^ to P : time from A to 
 
 =AP:A0=BQ:BT 
 
 = CD : CT by similar triangles. 
 
 COE. 1. Since the cifcle with centre and radius OD 
 will in general cut CT in two points, there -N^rill in gefieral be 
 two periods at which the particle^ are at a distance from eadh 
 other equal to OD ; we leave it as an exercise for the student 
 to form the construction for the other position. 
 
 Cor. 2. Since OD cannot be less than a certain dis- 
 tance, viz. the perpendicular from to CT (unless T and 
 coincide) we see that the particles will approach each other 
 till their distance is equal to this perpendicular, and is then 
 a minimum, and afterwards they will recede from each other. 
 
 If P, Q be the centres of two spherical balls, the proposition 
 will enable us to examine the circuhistancfes oi. theit" approach 
 to each other &c.; — if the distance PQ = sa.m of the radii of 
 the balls, we find when and where they will come into con- 
 
UNIFORM MOTION, 183 
 
 tact — ^if the sum of their radii be < the perpendicular from 
 to GT, we can find when and where they are nearest to 
 each other. 
 
 51. An analiftical solution of the above problem may be 
 given asfoUovis. 
 
 Let a, b be the co-ordinates of one particle A, and a', b' of 
 the other B, at first ; u, v the velocities of A estimated in 
 direction of the axes of a;, ^^ ; w, v the corresponding velocities 
 of the other J3; then after an interval of time * the co- 
 ordinates of 
 
 A will be a + uf, b '+ vt, and of 
 
 B a'+u't, b' + v't, 
 
 and if S be their distance at this time we must have 
 
 S== [a-a +{u- u') . tf+ [b-b' + {v- v) . t]\ 
 
 an ecfuation for determining t, the time when the distance 
 between the particles is S. This equation has two roots, from 
 which we may draw the same conclusion as in Cor. 1, Art. 50. 
 
 We may arrange the equation in the form 
 
 ^ = E-Wt-\-Ce, 
 
 in which E, D, G do not involve t, but only the known 
 quantities a, d, b, b', u, u', v, v', 
 
 or S' = —^ '- ; 
 
 from which we see that as EG — B^ is. essentially positive, 
 S will be least when Ct — B = 0, which corresponds to the 
 case of Cor. 2. 
 
 - 52, We proceed to discuss the problem of the collision 
 of two bodies. 
 
184 or UNIFOEM MOTION AND COLLISION. 
 
 All bodies ■with, which we are acquainted are capable 
 of being compressed more or less, and have a tendency in 
 different degrees to recover their original forms when the 
 compressing force is removed. This property we call their 
 elasticity: and the internal force which any body exerts to 
 recover its original form is called the force of restitution. 
 
 The ratio which the force of restitution bears to the force 
 of compression is found by experiment to be the same for the 
 same substance, whatever be the amount of the compressing 
 force, but to be different for different substances. This ratio, 
 which is generally represented by the symbol e, is taken as 
 the measure of the elasticity of any substance, and is fre- 
 quently called the modulus of elasticity. 
 
 This modulus can in no case be greater than unity; those 
 substances for which it is equal to unity are said to be 
 perfectly elastic, all others are imperfectly elastic ; and the 
 greater the numerical value of this modulus, the greater do we 
 regard the elasticity of the substances we are comparing. 
 
 Probably no substances are perfectly elastic; in steel 
 balls the value of the modulus e is about |, in glass about -if. 
 
 For the results of experiments on the elasticity of bodies 
 see Re^ports of the British Association for the Advancement of 
 Science, Vol. III. 
 
 53. In considering the effects of collision we shall sup- 
 pose the bodies to be spheres, perfectly smooth, and of uni- 
 form density, — so that their centres of gravity coincide with 
 their geometrical centres. 
 
 Def. The line joining the centres of the spheres at the 
 instant of impact is called the line of impact; when the centres 
 are moving in the line of impact, the impact is said to be 
 direct, — and in all other cases oblique. 
 
ELASTICITY. 185 
 
 Wlien a ball A impinges directly on a ball B, tbe effect 
 of the mutual pressure between them will be to accelerate 
 B and retard A ; and this will continue till their velocities 
 become equal. When the velocities are equal the mutual 
 pressure between them will cease, if the balls are inelastic, 
 and the balls will move on together uniformly with the 
 velocity which they then possess. 
 
 The intensity of this mutual pressure will vary during 
 the short time the balls press against each other ; but so far 
 as its effect in producing momentum is concerned we may 
 regard it as retaining some mean uniform value (see Art. 18), 
 and we may measure the effect of the collision, by the mo- 
 mentum X gained by B and lost by A : these effects on A 
 and B being equal in magnitude and opposite in direction 
 (Art. 46). 
 
 54. If the balls are elastic, the mutual pressure between 
 them will coiitinue after their velocities have become equal, 
 in consequence of the efforts they make to recover their 
 original forms; and the momentum gained by one and lost 
 by the other after that time (which we may call X') will bear 
 to the momentum generated during the first part of the 
 collision a ratio {e : 1) depending upon the, elasticity of the 
 substances : so that X' = eX, and the whole momentum gained 
 by B and lost by A will be expressed by X+X' or (1 + e) X. 
 
 The time during which this entire action is performed is 
 too small to be appreciated, but the illustration we have given 
 may serve to render the nature of it more intelligible, and 
 convey an idea of what is meant when it is said that Impact 
 is a pressure of short duration. 
 
186 OF UNIFORM MOTION AND COLLISION. 
 
 55. Two melasiio halls moving, with given velocities irn^ 
 pinge directly wpon each other; to find the vehedty of each after 
 impact. 
 
 Let u, V be the velocities of the two balls A and B 
 respectively, before impact ; and let ^_^^ 
 
 the direction of the arrows indicate I / j ^/''«^ 
 the direction of motion. Since they °i V yv_y 
 are inelastic, they will move on 
 
 together in the same direction after the impact with some 
 common velocity, which we may call u', — ^let X be the 
 momentum lost by J. and gained by5 dm-ing the impact; 
 
 then Au' — momentum of A after the impact, 
 = momentum before impact — X, 
 
 = Au-X. (i), 
 
 Bu = Bv + Xhj similar reasoning (ii) ; 
 
 ,-, adding {A + B)u' = Au + Bv (iii), 
 
 , Au + Bv ,. , 
 
 ^^"=^rF:r ^^^)' 
 
 and substituting this in (i), 
 
 X=Au-Au' = A{u-u') 
 
 t ( Au + Bv\ AB (u-v) , , 
 \ A+B J A+B ^ ' 
 
 Equation (iv) gives the common velocity of each ball 
 after the impact, and (v) gives the momentum gained by B 
 and lost by A. 
 
 COE. 1. We see from (iii) that the whole momentum of the 
 two balls is the same after impact as before it, — a result we 
 might have anticipated from the principle of Art. 46. 
 
dieeCt impact. • 187 
 
 Gob. 2, We may put the results in the foliowmg form : 
 
 velocity lost hjA = u-u' = ~== B{u-v) 
 
 A A+B ' 
 
 velocity gained }sjB^u'-v=^= ^ ^" ~ ^^ ; 
 
 in which shape they are sometimes useful. 
 
 05s. If B be tndving in a direction opposite to that of A 
 before impact, W6 have only to cha;nge the sign of v in the 
 above investigation : in other words, we may regard u, v 
 as representing the velocities of ^, B algebraically, — ^the proper 
 signs being given to them in any particular example in 
 accordance with the actual directions of motion. 
 
 The same remark applies to the Subsequent propositions of 
 this chapter. 
 
 CoE. 3. I^ A impinges on B at rest, we have simply 
 to put « = in the above results. 
 
 56. Two imperfectly elastic halls moving with given velo- 
 cities imping^ directly upon each other/ to find ike Velocity of each 
 after impacts (See fig. Art^ 55.) 
 
 Let u, u be the velocity of A before and after the impact ; 
 V, v' the same with respect to B ; 
 
 and let A, B represent the masses of the balls, the direction of 
 their motion being indicated by the arrows in the figure. 
 
 Let A impinge upon B, and let X be the momentum lost 
 by the former and gained by the latter during the first part 
 of the impact, i.e. before their velocities become equal, gene- 
 rated by the force of compression; and X' the momentum gene- 
 rated by the force of restitution, after the Velocities have 
 become equalj and which causes the balls to separate. 
 
188 OF UNIFORM MOTION AND COLLISION. 
 
 Then if e be the modulus of elasticity X' = eX, and X+X' 
 or (1+ e) X is the whole momentum lost by A and gained 
 by B, 
 
 whence Au = Au — {1+e) X\ ... 
 
 Bv' = Bv + {l+e)Xj' ^''' 
 
 Now X, being the momentum generated by the mutual 
 pressure of the balls before their elasticity comes into play, is 
 the same in magnitude as if the balls were inelastic, and 
 
 therefore, by the previous proposition X = — . ., . 
 Hence, substituting in (i), 
 
 .' = . + ii+e)§=^v + il + e)4^y 
 
 These two equations give the velocities of A and B after 
 impact. 
 
 Cor. 1. By adding equations (i) we get Au' + Bv = Au 
 + Bv; that is, the whole momentum is unchanged by the 
 impact. 
 
 CoE. 2. We may put the results expressed by (ii) in the 
 form 
 
 velocity lost by ^ = m — m' = (1 + e) —^ ^ 
 
 J. ...(iii). 
 velocity gained hy B = v' — v = {1 +e) — j — ^ 
 
 Also from (ii) we get by subtraction 
 
 v' — u'=v — u + {l+e){u — v)=e.{u-v) (iv) ; 
 
OBLIQUE IMPACT. 189 
 
 i.e. the relative velocity of -4 and B after impact: their 
 relative velocity before impact = v —tl i u — v = e il. 
 
 COE. 3. If A impinges upon B at rest, we have simply 
 to put w = in the above results. 
 
 57. The problem of diirect collision of two balls (Art. 56), 
 is sometimes solved by assuming (i) that the total momentum 
 after impact is the same as before impact, — this would follow 
 from the principle that action and reaction are equal and 
 opposite ; — and (ii) that the relative velocity of the two balls 
 after impact bears a constant ratio to their relative velocity 
 before impact, say the ratio e : 1 ; — this result being the state- 
 ment of an experimental fact, and e being then defined to be 
 the modulus of elasticity. 
 
 On these two assumptions we should have, with the 
 notation of the preceding article, 
 
 Au' + Bv =Au + Bv\ 
 and v —u =e[u — v) J ' 
 from which we readily obtain the results marked (ii) in 
 Art. (56), or we may obtain u v in the equivalent forms 
 
 Au + Bv B_ 
 " A + B ^' A 
 
 u — v)\ 
 
 , Au + Bv , A(u — v) 
 
 Obs. We prefer the mode of treating the problem given 
 in Art. (56), as it is more strictly referred to the simple laws 
 of motion than the method of this article. 
 
 58. Two smooth imperfectly elastic balls, moving in one 
 plane with given velocities in given directions, impinge obliquely 
 an each other, — to determine the motion of each after impact. 
 
190 OF UNIFORM MOTION AND COLLISION. 
 
 Xiet Ox be the line passing through -the centres of the 
 balls at the instant of impact, 
 and let the arrows indicate the 
 direction of motion of the ball? 
 before and after impact. 
 
 Let w, u be the velocities of A 
 before and after impact in directions 
 making angles a, 6 with Ox, 
 
 V, v, ^, <j) similar quantities with respect to B. 
 
 Now, since the ba,lls are smooth the mutual action between 
 them will take place entirely in the direction Ox, and hence 
 it will be convenient to estimate the velopities of the bajls in 
 direction of Ox (Art. 20), and at right angles to Ox; and 
 these motions may, by the second law of motion, be treated 
 separately. Since no force acts on either ball perpendicular 
 to Ox their velocities resolved at right angles to Ox will . 
 remain unchanged by the impact, whence we get 
 
 M'sin ^ = Msina"l (i), 
 
 v'sm(j) = vsm^ ) (ii), 
 
 and farther, the resolved velocities in direction Ox are aifected 
 by the impact just to the same extent as if these resolved 
 velocities alone existed. 
 
 ' , a\' ' J. I being the velocities of A, B 
 u cos o ) V cos (pj° ' 
 
 resolved along Ox, „ !• the impactj'^-if X be the momen- 
 tum gained by B and lost by A during the impact, we should 
 get as in Art. (56), 
 
 „ (1 + e) AB (m cos a rr- V cos yS) 
 
 ^ zt:b ' 
 
OBLIQUE IMPACT. 
 
 191 
 
 and we get 
 
 B 
 
 'cos0 = MCOsa-(l + e) . „ (w cos a - y cos ^) i...(iii), 
 
 t)' cos ^ = ■?; cos /S + (1 + e) - 
 
 , (m cos a — ^7 cos /8) \ . . . (iv). 
 
 ' A + B'' 
 
 The equations (i), (iii) suffice to determine u and €, and 
 (ii), (iv) to determine v and ^: — and tliese four quantities 
 define the magnitude and direction of the velocities of the 
 two balls. 
 
 Ohs. The a'bove expresses in general terms the solution 
 of the problem of the collision of two balls moving in one 
 plane ; any particular cases can of course be deduced from it, 
 by assigning to the symbols involved their proper values, and 
 this the student can readily do for himself. We will only 
 notice the following interesting case. 
 
 COE. If a ball A impinge obliquely upon a very large 
 ball B at rest, we have 
 
 A . . B 
 
 ^ = ^' AVB 
 so that we get 
 
 u sin 6= M sin 
 u cos 6 = — eu cos 
 and v = 0, very nearly. 
 This shews that the motion which is 
 communicated to B is inappreciable; and 
 since cot^ — — e cot a we must have 6 > 90°, 
 and therefore A is reflected: — a ball strik- 
 ing a Jixed plane is a. case of this kind; 
 or again, when a ball strikes ;the earth, the 
 mass of the earth is so great compared with 
 that of the ball that the motion communi- 
 cated to it is insensible. 
 
 = 0; and ^ ^ =1, very nearly, 
 j-, "which .give w',and 0, 
 
192 
 
 OP UNIFORM MOTION AND COLLISION. 
 
 59. When two balls impinge upon one another and their 
 motion does not take place in one plane, in order to de- 
 termine the subsequent motion of the balls we must employ 
 the same principles as those we have used in Art. (58), viz. 
 resolve the velocities of the balls in the direction of impact 
 and at right angles to it : the latter will be unaffected by the 
 impact, and the former will be altered in the same manner 
 as if they alone existed. The formulae which express the 
 general solution of the problem require a knowledge of Geo~ 
 metry of three dimensions, and are too complicated to be given 
 here. 
 
 60. A tall impinges obliquely upon a fixed smooth plane; 
 to find the motion of the hall after impact. 
 
 Let PQ be the normal to the plane at the point where the 
 ball is in contact at the instant of 
 impact: let the plane of the paper 
 contain this normal, as well as the 
 line of -4's motion before impact, and 
 intersect the fixed plane in the line 
 GPD ; then the line of ^'s motion 
 after impact will lie in this same 
 
 plane, since no force acts on the ball during the impapt at 
 right angles to this plane. 
 
 Let a, 6 be the inclination to PQ of J.'s velocities before 
 and after the impact ; u, u the velocity of A before and after 
 impact, X the momentum destroyed by the force of compres- 
 sion. Then the velocity parallel to CD being unaffected by 
 the impact, we have 
 
 w'sin d = u&iaa. (i); 
 
MOTION OF CENTRE OF GRAVITY. 
 
 193 
 
 (iii). 
 
 and since the momentum of A resolved along the normal Q'B 
 is entirely destroyed by the plane 
 
 X= Au cos a, 
 
 and eX is the additional momentum generated in the opposite 
 direction by the elasticity, or force of restitution ; 
 
 ••. .eX=^M'cos Q, 
 
 whence, m' cos 5 = ew cos a (ii).- 
 
 From (i) and (ii) we get 
 
 cot = e cot a 
 
 m' = M VCsin" a + e" cos" a)J 
 These equations (iii) determine the velocity and direction 
 of motion of A after impact. 
 
 Obs. The student may compare this solution with the 
 solution of what is substantially the same problem, deduced 
 in Art. (58), Cor, 
 
 Cor. 1. If the ball be inelastic,- e = 0; whence ^=90°, 
 and m' = M sin a ; i. e. when an inelastic ball impinges obliquely 
 on a fixed plane, after impact it will move along the plane with 
 a velocity equal to u sin a. 
 
 Cor. 2. The impulse sustained by the plane will be 
 = A (MCOsa + M'cos^) = (l + e)^MCOsa. 
 
 61. To find the velocity of the centre of gravity of two halls 
 moving uniformly in one plane. 
 
 Let the position and motion 
 of the two balls be referred to the 
 two rectangular axes Ox, Oy in 
 the plane in which they move, 
 which we may suppose to be the 
 plane of the paper. 
 
 p. M. 13 
 
 y 
 
 B 
 
 1 
 
 — 
 
 n 
 
 1 
 
 ■'{ 
 
 a) 
 
 
 
 
 
194 OF UNIFORM MOTION AND COLLISION. 
 
 Let A, B be the centres of the balls at first, 
 
 A', B' after an interval t, 
 
 a, h co-ordinates of A, B measured along Ox, 
 X, X A , B' 
 
 M, u the velocities of A and B resolved parallel to Ox, 
 which will be uniform, since the balls. are supposed to move 
 uniformly (Art. 20), 
 then x—a-\-ut 
 
 x'^h + u'tl ^'^" 
 
 and if x, x be the co-ordinates of G the centre of gravity of 
 A and B in the first and second positions of A and B, measured 
 along Ox, — we have by Statics, Art. 74, 
 
 {A+B)x = Aa + Bl \ .... 
 
 {A+B)x' = Ax-^Bx] ^''^' 
 
 .-. {A + B)[x -'^ = A{x-a)+B{x' -h) 
 = {Au + Bu')t; 
 _, _ Au + Bu , ,.... 
 
 .-. X -x= ji + B ~ '"^^' 
 
 Now this represents the space passed over by G, measured 
 parallel to Ox, — and it cc i the time — consequently the velo- 
 city of G parallel to Ox is uniform and 
 
 Au + Bu' - 
 = ^^j^ = u suppose. 
 
 Similarly, if v, v, v be the velocities of A, B, G parallel 
 to Oy we should have 
 
 _ Av + Bv 
 '"^'AVB-' 
 
 whence u, v being known, the motion of G is known. 
 
MOTION OF CENTRE OP GEAVITY. 195 
 
 CoK. 1. If there were three or more balls, by a similar 
 process we should obtain 
 
 - Au + Bu'+Cu"+... t(Au) 
 A + B+G+... ~ XiA) ' 
 
 -_ Av + Bv'+ Gv"+... 2{Av) 
 "" A + B-\-G + ... %{A) ' 
 
 and if the motions of the balls were not confined to one plane, 
 and we introduced a third co-ordinate axis at right angles to 
 Ox and Oy, and represented the velocities parallel to this 
 axis by w, w, w'.-.we should have 
 
 _ _ Aw + Bw'+ Cw" + ... _ t {Aw) 
 ^~ A + B+V+... ~X.{A) ■ 
 
 These formulae are analogous to those for the position of 
 the centre of gravity of a system of bodies {Statics, Art. 74). 
 They may. be expressed generally thus : The velocity/ of the 
 centre of gravity of a system of hodies estimated in a given 
 direction is equal to the sum of the momenta of the several 
 hodies estimated in the same direction, divided hy the mass of 
 the system. Or, if each hody of a system ie moving uniformly, 
 the centre of gravity of the system also moves uniformly with a 
 velocity such that the whole momentum of the system estimated 
 in any given direction is equal to that of a single hody {equal in 
 mass to that of the system) coincident with the centre of gravity, 
 and moving with the same velocity as the centre of gravity, 
 
 N.B, The acceleration of the centre of gravity would be 
 obtained by formulae exactly similar to those obtained above 
 for the velocity — the accelerations of the several bodies being 
 written in the formulae instead of their velocities. 
 
 Cob, 2. Since it appears by Art. 40 that if we impress 
 any the same velocity upon each body of a system, the relative 
 
 13—2 
 
196 OF UNIFORM MOTION AND COLLISION. 
 
 motions of the parts of the system are not affected thereby, — 
 
 suppose we wish to reduce the centre of gravity of two balls 
 
 to rest by impressing velocities equal to —u, —v...on each 
 
 ball, we see that the momentum to be communicated to 
 
 A, B for this purpose would be 
 
 T^ . Au + Bu „ Au + Bu 
 
 -Au,-Bu...ox-A. ^^-^ , -B. ^^^ 
 
 parallel to Ox ; and — Av, — Bv parallel to Oi/. 
 
 62. When two smooth halls im/pinge upon one another the 
 motion of the centre of gravity is unaltered by the impact. 
 
 First, let the balls be moving in the line of impact Ox, 
 i. e. let the impact be direct (fig. Art. 55), 
 
 ' , \ velocities of „ | before and after impact, 
 V, V I B) 
 
 u, u velocity of the centre of gravity before and after 
 impact ; 
 
 , - Au + Bv — Au' + Bv' 
 then«=^-|-g-, u= ^^^ ; 
 
 and (Art. 56, Cor. 1) the whole momentum is the same after 
 impact as before, therefore Au + Bv = Au + Bv'; whence we 
 get u = u', which proves the proposition. 
 Secondly, let the impact be oblique. 
 
 Resolve the velocity of each ball in direction of impact 
 and at right angles to it ; by the first case the velocity of the 
 centre of gravity in direction of impact will be unaltered; 
 and since the velocity of each ball resolved at right angles to 
 the direction of impact is unaffected by the impact, the 
 velocity of the centre of gravity in this direction will not be 
 changed by the impact, — consequently the velocity and direc- 
 
EXAMPLES. 197 
 
 tion of motion of the centre of gravity of the balls are the same 
 after impact as before. 
 
 CoE. We can without much diflSculty extend the theo- 
 rem of this article to the case of several balls, and shew that 
 " the motion of the centre of gravity of any number of smooth 
 balls is not changed by the impact inter se of two or more 
 balls of the system." 
 
 Examples and Problems. 
 
 63. (I) A ball of dibs, weight moving from left to right, 
 with a velocity of 8 yards per second, impinges directly upon 
 a ball of 10 lbs. weight moving in the same direction with 
 a velocity of 2 yards per second; determine their motion after 
 the impact. 
 
 (i) When the balls are inelastic. (Art. 55.) 
 
 Since the weights of the balls are in the ratio of their 
 masses, we may take 4 and 10 to represent their masses, and 
 we shall have 
 
 their common velocity after impact = 
 
 ^ 4 + 10 = 14 " ^^ ^^^^® P^^ ^^'^^^^ ' 
 
 ,_ AB{u-v) 4.10.(8-2) 240 ,,, 
 '^^^= A + B = 4 + 10 =T4=^'^' 
 
 i. e. the mutual pressure between the balls is capable of gene- 
 rating a velocity of 17f yards per second in a mass whose 
 weight is 1 lb. 
 
 (ii) If the balls are elastic, then using the same notation 
 as in Art. (56), 
 
198 OF UNIFORM MOTION AND COLLISION. 
 
 velocity of A after impact = 8 — — :^ = — =- e, 
 
 B ^ (l + .)4(8-2) ^2_6 12 
 
 ^ 4+10 7 7' 
 
 a.ndX=17f(l+e). 
 
 13 
 
 If e = — , the ball A will be at rest after the impact ; 
 
 xO 
 
 13 
 
 and according as e < or > -v , A will follow B with a less 
 
 velocity or be reflected back and move in the opposite 
 direction. 
 
 64. (II) A ball A moving with a given velocity impinges 
 directly upon a ball £ at rest, and B afterwards impinges 
 directly upon a ball G at rest; find the velocity communicated 
 to 0. 
 
 If u be the original velocity of A, we have by Art. (56), 
 velocity of B after first impact = -^ — -^ u = v suppose, 
 
 ^ -^ f n e. ■ X (1 + e) -B (1 + eYAB 
 
 velocity of C after impact = i^ . = ^/^^^[^^fj^ u. 
 
 Cor. 1. The velocity communicated to G by the interven- 
 tion of B will vary with the magnitude of B, and will be the 
 
 greatest possible when , . „ is greatest ; 
 
 i. e. when ^ ^ is least, 
 
 and since this may be written in the form 
 
 {v(^)-^(^)}V{v(^)+v(c)r, 
 
IMPACT ON A EOUGH PLANE. 
 
 199 
 
 this will be the case when B=i\/{AC); — in other words, the 
 velocity of G will be greatest when 5 is a mean proportional 
 between A and 0. 
 
 65. (Ill) A particle is to be projected from a given point 
 P so as to pass through another 
 given point Q, after being reflected 
 at a given fixed plane AB; to 
 find the direction of projection. 
 
 Suppose T to be the point 
 where the particle must strike the 
 plane, then the plane PTQ must 
 be perpendicular to the fixed plane, and will cut it in a straight 
 line AB. 
 
 Now the particle impinging on the plane in direction PT 
 and being reflected in direction TQ, we must have 
 
 tan QTS=e.ta.nPTA (i) Art. (60). 
 
 If Q8 be drawn perpendicular to AB, and P 7 produced to 
 meet ^;S' in B, we shall have 
 
 tanQTS=eta,nBT8, 
 
 and therefore QS = e . SB. 
 
 This si^gests the following simple construction for deter- 
 mining T. Draw Q8 perpendicular to AB and produce it to 
 
 B, making SB=-. Q 8; join PB cutting AB in T. Then 
 
 the condition (i) is satisfied, and PT is the direction in which 
 the particle must be projected. 
 
 CoE. If the particle is to pass through Q after reflexion 
 at two planes TV, US in succession, we have the following 
 
200 
 
 OF UNIFORM MOTION AND COLLISION. 
 
 construction. Draw Q8R perpendicular to the latter plane, 
 making 
 
 8B=-. Q8. 
 e 
 
 Draw BVD perpendicular to the 
 first plane, making 
 
 Vn^-.BV; 
 e 
 
 join FD cutting the first plane in T, — join TB cutting the 
 second plane in U, — then if the particle be projected in direc- 
 tion PT it will be reflected along TV and again reflected at U 
 in direction UQ, and so pass through the point Q. 
 
 66. (IV) A heavy particle impinges upon a fixed rough 
 plane ; to find its motion after impact. 
 
 Let the plane of the paper represent the plane of impact, 
 i.e. the plane which contains the 
 direction of motion of the particle 
 before impact, and the normal to 
 the fixed plane at the point of 
 contact. 
 
 Let u,.u' be the velocities of 
 the particle (mass A) before and after impact. 
 
 a, 6 the angles its direction of motion makes with the 
 normal QN before and after impact. 
 
 X, F the momentum generated by the fixed plane in the 
 particle, in directions §^and QG, — the latter arising from the 
 roughness of the plane. 
 
 Then resolving the motion in direction QN and CD, we 
 have, as in Ait. (60), 
 
IMPACT ON A ROUGH PLANE. 201 
 
 m'cos d = eu COS a , (i), 
 
 the complete value of X= (1 + e) -4m cos a (ii), 
 
 and Au' sin ^ = -4m sin a — i'' (iii). 
 
 Now we may take F = fiX {iy), where fi depends upon 
 the roughness of the plane, and is a numerical quantity to be 
 determined by experiment, it is sometimes called the coefficient 
 of dynamical fnction ; from these four equations we get 
 
 u' cos = eu cos a | 
 
 u' sin ^ = M sin a — /* (1 + e) M cos a j ' 
 
 which two equations determine u' and 6, i. e. the velocity and 
 direction of motion after impact. 
 
( 202 ) 
 CHAPTER III. 
 
 OF UNIFORMLY ACCELERATED MOTION. 
 
 67. The accelerating force upon a particle is said to Le 
 uniform when equal increments of velocity are added in equal 
 increments of time, however large or small- these increments 
 of time may be. 
 
 Hence, in accordance with the definitions and conventions 
 of Arts. 5, 13, if v be the velocity of a particle at the end of a 
 time t, during which it has been subject to a uniform accele- 
 rating force f, and if u were its velocity to begin with, we 
 shall have_/. t to represent the increment of velocity, and 
 v=u+ft (i). 
 
 If the particle started from rest u = and v =Ji. 
 
 Obs. The formula (i) is algebraically true in any case 
 where the force is really a retarding force (Art. 6, Obs.), or 
 where the velocity u at the beginning of the time t exists in 
 a direction opposite to that in which v is measured : in any 
 case it is necessary simply to assign the proper algebraic sign 
 to u andy) and the result (i) will be available. 
 
 68. If s be the space described from rest in time tby a 
 particle under the action of a uniform accelerating force f then 
 will s = \ff. 
 
 Let the time t be subdivided into n intervals, each equal 
 to T, so that WT = < ; then the velocities at the beginning 
 of the 
 
 1st 2nd 3rd wth of these intervals of t 
 
 will be /t 2/t... {n-\)fr; 
 
UNIFOKM ACCELERATION. 203 
 
 and at the end of the same intervals will be 
 
 JT, 2/T, ZfT...nfr. 
 
 Now if the particle were to move during each successive 
 interval of t, with the velocity which has at the beginning 
 of that interval, the space described would be 
 
 = . T+/r.T + 2/T . T + ... + (n- l)yT. T, 
 which is 
 
 =yV» {1 + 2 +...+ («-!)} 
 
 « (m — 1) 
 
 1.2 
 
 ^A^4/^^(i-J),Bi 
 
 smce m ■■ 
 
 And again, if the particle were to move during each 
 successive interval, with the velocity which it has at the end 
 of that interval, the space described would be 
 
 =yT.T+2/T.T+3/r.T+...+«/T.T, 
 
 which is 
 
 =yT»(i + 2 + ...+«)=«^/i-^=l/«^(i+i). 
 
 Since the velocity is continually increasing during the 
 time t, the space actually described by the particle will be 
 intermediate to the spaces described under these two hypo- 
 theses, i.e. 
 
 s lies between "^fe {l - i) and ^/if [l + ^) 
 
 however large n be taken ; but when n is taken indefinitely 
 large, these two limits each become \f^, and therefore s 
 which always lies between them must coincide with them in 
 the limit, that is s = \f^; and if v be the velocity at time t 
 we hate v =ft and thence v' = 2fsi 
 

 
 y" 
 
 
 
 ^ 
 
 
 
 tl 
 
 y' 
 
 
 
 J 
 
 /- 
 
 
 h 
 
 y^ 
 
 
 L^ 
 
 
 A B C H 
 
 
 K 
 
 204 OP UNIFORMLT ACCELEEATKD MOTION. 
 
 69. The same result (s = i-j^) may be arrived at very 
 simply by the following geome- 
 trical process. 
 
 Let the straight line AK re- 
 present the time (s) of motion 
 from rest, and let this be divided 
 into n equal parts AJB, BO, CD. . . 
 let the lines Bb, Gc,...KT drawn 
 at right angles to AK represent 
 
 the velocities acquired at the end of the successive intervals ; 
 the points b, c ... T will lie in a straight line, since the velo- 
 city varies as the time from rest. Complete the inner and 
 outer series of parallelograms, as in the figure. 
 
 Now if the particle be supposed to move uniformly during 
 any interval (as GB) with the velocity Cc which it has at the 
 beginning of that interval, the number of units of area in the 
 parallelogram cB will represent the number of units of length 
 passed over by the particle during that interval. With this 
 understanding, the sum of the inner or outer series of paral- 
 lelograms will represent the space passed over by the particle, 
 supposing it to move during each interval with the velocity 
 which it has at the beginning or end of that interval respectively; 
 and the actual space described lies between the spaces de- 
 scribed on these two several suppositions. But when the 
 number of intervals is increased, and their magnitude dimi- 
 nished indefinitely, each series of parallelograms approximates 
 to the triangular area AKT, which will represent the actual 
 space described by the particle ; and since 
 
 AK= t, and ^7=^ = velocity at time t, 
 
 .-. s = ^AK.KT=^t.ft = ^ff. 
 
 Obs. The student will remark that the above is sub- 
 
OF UNIFORMLY ACCELERATED MOTION. 205 
 
 stantially a geometrical illustration of the proof of the pro- 
 position given in Art. 68. 
 
 70. A particle is projected with velocity u, and acted upon 
 hy an accelerating force fin the direction of motion. To find 
 the relation between the space (s) passed over, the time (t) of 
 motion and velocity (v) acquired. *' 
 
 The particle at any time is moving with a certain velocity, 
 and so far as the subsequent motion is concerned it is imma- 
 terial how we suppose that velocity to have been acquired. 
 Let -then the force _/ generate a velocity u by acting for a time 
 i' and through a space s', then we have u=ft' ; and if the 
 particle continues subject to the action of the same force, and 
 passes over a space s in time t, we have s + s' described /rom 
 rest in time t-\-t'; 
 
 .: s + s' = ^f{t + t'Y and s' = \ft'^ ; 
 .: s = ^f{f + 'itf)=ut + lfe (i); 
 
 also v=f{t' + t)=U+ft (ii); 
 
 whence also 
 
 «2 = {u +ftf = m' + ifiut + \fe) = m" + 2/s . . . (iii) . 
 Equations (i), (ii) express the relations required. 
 
 Ois. If the force act in a direction opposite to that in 
 which s, u, and v are estimated positive, we must change the 
 sign of/ in (i), (ii), (iii), and we get 
 
 s = ut-\ft\ v = u-ft, v' = u^-2fs (iv). 
 
 The student will have little difficulty in obtaining any of 
 the results of (i), (ii), (iii), (iv), of this article, by a geometrical 
 proof similar to that in Art. 69. 
 
 71. We may arrive at the same results thus by an appli- 
 cation of the principle stated in Art. 40. 
 
 A' A p 
 
206 OF UNIFOKMLY ACCELERATED MOTION. 
 
 Let the particle be projected from A in direction AP with 
 the velocity u ; — the relative motion of A and P will he the 
 same if we impress upon hoih a velocity equal to u in, the 
 opposite direction ; this reduces P to initial rest, and if P, A 
 be simultaneous positions of the particle and oi A, v their 
 relative velocity at that time t, and A P=s, we have 
 
 AP=\fe AA' = ut (i); 
 
 .-. s=ut + ^fe.. (ii), 
 
 and v = u +ft ; 
 the same results as before. 
 
 72. Note. The same results might have been arrived at 
 by a process similar to those employed in Art. 69. These 
 we leave as an exercise for the student. 
 
 We would here caution him likewise against a loose and 
 incorrect application of the second law of motion to this 
 problem which we have noticed in some works on dynamics. 
 They state that the space described in consequence of the 
 initial velocity is = ut, and the space that would be described 
 in the same time by the action of the accelerating force y is 
 = \f^, and therefore hy the second law of motion, the whole 
 space described is the sum of these two, or s = m< + \jf. The 
 result arrived at is true, but the principle assumed is unsound, 
 for the second law of motion states the theory of the action of 
 forces at & particular instant, and asserts nothitig directly as to 
 the quantitative effects accruing in &nj finite time. 
 
 73. When a particle starts from rest (Art. 68) 
 
 V =ft and s = \ff ; 
 
 and from these two equations a relation can be obtained 
 between any three of the quantities v, s, f t. 
 
GENERAL FOEMULiE. 
 
 207 
 
 Th 
 
 US 
 
 -fi=j=^{m 
 
 /_?'- h! _ ?£ 
 
 w 
 
 2s 
 f 
 
 And again, if tHe particle start witli the velocity u 
 v = u +ft, 
 
 v' = u' + 2fs; 
 forms which it is desirable the student should rememlDer. 
 
 CoE. The space described in t" from rest = kft'', 
 
 (<-i)" =if{t-iT; 
 
 .•. space described during the i* second = \f{2t — 1). 
 
 Hence the spaces described during the 1st, 2nd, 3rd,... 
 seconds are \f. 1, \f. 3, \f. 5, ... &c., and are in the ratio 
 of the consecutive odd numbers, 1, 3, 5... 
 
 The -result s = —r — t shews that the space described in 
 
 any time is the same as if the particle had moved uniformly 
 during the whole time with the mean velocity. 
 
 74. One of the simple cases of a uniform force is that 
 of gravity, the accelerating effect of which is uniform. (Art. 
 43.) We give an example of the application of the preceding 
 results to this case. 
 
203 OF UNIFOEMLT ACCELEEATED MOTION. 
 
 Ex. A particle is projected vertically upwards with a velo- 
 city oflOQ feet per second, to find (i) its height at the end of Z", 
 and (ii) the time when it is at a height of 140 feet above the 
 point of projection. 
 
 When a. foot and a second are taken as the units of space 
 and time the numerical value of ^ = 32.2, (Art. 43), and if u 
 be the velocity of projection, and s the height at time t after 
 projection, we have s = ut — \gf, (Art. 70). 
 
 For the first part of the example i = 3, u= 100 ; 
 .-. s = 3.100 - 1 32.2 . (3)" = 155.1 feet 
 
 = height of the particle at the end of 3 seconds. 
 
 For the second part of the example s = 140, u = 100 ; and 
 we have to find t from the quadratic equation 
 
 s = Mi — \gV. 
 
 Solving the equation we get 
 
 9 
 
 Substituting the numerical values of u, g, s, we get after 
 reduction, 
 
 a double result, which is to be explained thus, — at the end of 
 2". 13 the particle is at a height 140 feet in its ascent, and at 
 the end of 4".08 it is again at the same height of 140 feet on 
 its descent, after having reached its highest point and then 
 descending. 
 
 75. We subjoin a few interesting problems which can be 
 solved by the principles already explained. 
 
PKOBLEMS. 
 
 209 
 
 r^ 
 
 Peob. Two bodies P, Q are connected hy an inextensible 
 string which passes over a smooth fixed pully ; 
 to determine the motion of each body, and the 
 tension of the string. 
 
 Let P, Q represent the masses of the bodies ; 
 Tthe tension of the string, the mass of which 
 we will neglect, and suppose P> Q. 
 
 Now moving force on P downwards = Pg — T, 
 Q upwards = T— Qg; 
 
 Pg-T^ 
 
 *. accelerating force onP downwards = 
 Q upwards = 
 
 P 
 T-Qg 
 
 (!)■ 
 
 Now the string being always stretched and inextensible, 
 the velocity of P downwards and of Q upwards will be always 
 equal, and therefore the rate of change of their velocities, 
 i.e. the acceleration of the two bodies must be equal; 
 
 Pg-J^^ T^ Qg 
 p 
 
 ^PQ 
 
 Q 
 
 whence T= 
 
 P+Q 
 
 9- 
 
 (ii), 
 
 which gives the tension of the string, — and further substituting 
 this value of T in either of the expressions (i), we get the 
 acceleration on P downwards and on Q upwards 
 
 T 
 ■9-15 = 9- 
 
 2Q 
 
 9 = 
 
 P-Q. 
 
 P+Q^ P+Q 
 
 -)9- 
 
 P-Q 
 
 Also velocity of P and Q after time t from rest =-p, — —at 
 
 \P- 
 
 . space described *~9 p j. o 
 
 r. M. 14 
 
 «..-. 
 
210 OP UNIFOEMLT ACCELERATED MOTION. 
 
 CoE. I. By taking F— Q a.s small as we please we may 
 make the motion as slow as we please, and so capable of being 
 measured, — by which means the value oi g might be obtained 
 from observation. This is substantially the principle of 
 Atwood's machine, which will be described hereafter, (Art. 82). 
 
 CoK. 2. If at any instant a jpart (B) of one of the bodies 
 ( Q for instance) were suddenly detached, there would be no 
 instantaneous change of the velocity of either body, but the 
 
 acceleration would become -r; — j=: — ^ff, and the tension of 
 the string would become „ ,, — ^- . g. 
 
 76. Peob. Two hodies P, Q are in motion, connected iy 
 a string which passes over a smooth fixed pully ; 
 another hody R is suddenly attached to Q; — -find 1)a 
 the change of velocity and the impulsive strain on 
 the string. 
 
 Let P, Q, B be the masses of the bodies, and 
 suppose B to become attached to § by a string J 
 connecting them suddenly becoming tight. Let V 
 be the velocity of P and Q at the instant before this takes 
 place, and V the common velocity of the three the instant 
 after, X^, X^ the impulsive strain on the strings PA Q, QB, 
 respectively; then for the motion of the three bodies we 
 have (Art. 46) 
 
 PV = PV-X^, 
 
 QV'=QV+X^-X„ 
 BV' = X,; 
 whence by adding (P+ Q + B)V'={P+Q) V, 
 
 -^' = p|jfl^^ «' 
 
 m. 
 
MOTION ON AN INCLINED PLANE. 211 
 
 andX. = P(F-F')=^p:ff^F. (ii), 
 
 (P+ Q) R „ ..... 
 
 ^^ = - p+Q + R ^ H. 
 
 Equations (i), (ii), (iii) determine the three quantities 
 required to determine the change of motion completely. 
 
 MOTION ON AN INCLINED PLANE. 
 
 77. Peob. a heavy hody Q is drawn wp a smooth inclined 
 •plane hy another hody P, which descends vertically ; P heing 
 connected with (J hy an inextensible string passing over the 
 vertex of theplane^ 
 
 LetP, 5 be the masses of the bodies, 
 T the tension of the string, and a 
 the inclination of the plane to the 
 horizon, R the pressure of Q on the 
 plane. 
 
 Then resolving the motion of Q parallel to the plane 
 and perpendicular to it, the weight of Q is equivalent to 
 a force Qg sin a down the plane, 
 
 Qg cos a. perpendicular to the plane, 
 
 the latter force is balanced by R, the pressure of the plane, 
 
 whence ^= ^^cosa (i), 
 
 and acceleration of Q up the plane = ^ , 
 
 .....P downwards = -^^ — . 
 
 And since the string continues stretched, the velocities of 
 
 14—2 
 
212 
 
 OP UNIFORMLY ACCELERATED MOTION. 
 
 P and Q in these two directions are always equal and there- 
 fore the accelerations upon them are equal, that is, 
 
 Pq-T T-Qffsina 
 F ~ Q ' 
 
 PQ (1 + sin a) g 
 
 whence T = 
 
 P+Q 
 
 ("), 
 
 and acceleration on P downwards and on Q up the plane 
 
 _ Pg-T P-QBma. 
 - P - P+Q 
 
 9 
 
 ,(iii), 
 
 equation (ii) gives the tension of the string, and (iii) gives 
 the acceleration from which the velocity acquired and space 
 passed over in any time may readily be obtained. 
 
 Cor. The preceding problem may be varied by sup- 
 posing Q to move on a smooth hori- 
 zontal table. The student may either 
 investigate the motion in this case inde- 
 pendently, or deduce the results from 
 the present Art. by making a = 0. 
 
 AP 
 
 78. A heavy hody descends freely down a smooth inclined 
 plane; — to find the time of motion and the velocity acquired. 
 
 Let P be the mass of the 
 body moving down the plane 
 BA, the inclination of which 
 to the horizon is a, R the 
 pressure on the plane. 
 
 Then resolving the forces 
 on P parallel to the plane and 
 perpendicular to it, 
 
MOTION ON AN INCLINED I^ANE. 213 
 
 we have moving force down the plane = Pg sin a, 
 
 perpendicular to the plane = Pg cos a, 
 
 and this latter force is counteracted by B, since there is no 
 motion perpendicular to the plane ; 
 
 .•. E = Pg cos a, which determines B, 
 and accelerating force down the plane =y sin a; 
 if t be the time of moving over BA from rest, and v the velo- 
 city acquired, 
 
 AB = ^^ sin a . <", v=g sina. .t. 
 
 Whence ^ = * / ( — '■ — ' ) > ^"<i v = \/{2g .s,ma. AB) 
 
 = ^{2g.BG). 
 
 The latter result shews that the velocity at A is the same 
 as that of a body falling freely through a vertical height equal 
 to BC, — that of the plane. 
 
 GoE. If the body start at B with a velocity u, and v be 
 its velocity after describing any length BA, and h be the 
 vertical depth of A below B, we shall readily obtain 
 
 79. A heavy particle is projected with given velocity up an 
 inclined plane- — to find its velocity at any point of its course. 
 
 We suppose the motion to take place in a vertical plane 
 BA G which is perpendicular to 
 the inclined plane. 
 
 Let the particle P be pro- 
 jected from A with velocity u 
 up the plane, v the velocity 
 after a time t when it has 
 described the space AP=s, 
 z the vertical height of P above 
 the horizon AC, a. the inclina- 
 
214 OP UNIFORMLY ACCELEEATED MOTION. 
 
 tion of the plane, so that a = s . sin a : the resolved force of 
 gravity down the plane tending to retard P is = ^ sin a. 
 
 Hence (Art. 70) v = u-gsina..t (i), 
 
 s = ut — \gBm.a..f (ii), 
 
 two equations connecting the three quantities v, s, t, so that 
 any one of them being given, the other two may be found : 
 these results are algebraically true if ii and s be one or both 
 negative. 
 
 If we eliminate t we obtain 
 
 v^=u'— 2g sina.. s = u'' — 2gz (iii), 
 
 a result which shews that the change of velocity can be ex- 
 pressed in terms of the vertical height through which the 
 particle has ascended — and is the same in amount as if the 
 particle had been moving freely in a vertical line upwards — 
 the time of motion however would not be the same in the 
 two cases. 
 
 From (ii) we can derive the value of t corresponding to 
 any value of s, viz. 
 
 t = —= 7-^ = -= — -. ^ (iv), 
 
 a double result, indicating the times at which the particle 
 will pass through the position F — {AP = s) — on its way up 
 and down the plane. 
 
 2 2 
 
 The maximum value of s is = ■. — and of « is =— . 
 
 2g sin a 2g 
 
 as may be seen from equations (iii) or (iv). 
 
 Ois. The results of this article will be* applicable to the 
 case of a particle projected freely vertically/ upwards, if we 
 write a = 90°, and .•. sin a = 1, 
 
QUICKEST DESCENT. 
 
 215 
 
 80. We may apply the results of the previous article to 
 the following proUem. 
 
 The time of descent of a particle down any cliord of a 
 vertical circle heginning at the highest point of the circle is 
 the same. 
 
 Let AB he the vertical diameter of 
 the circle, AP any chord drawn from A. 
 Then the accelerating force on the par- 
 ticle down AP = g cos PAB = g sin PBA, 
 
 2AP \ 
 
 and time down AP 
 2 . AB\ 
 
 VGi 
 
 V( 
 
 ')■■ 
 
 sin PBA) 
 since AP = AB sin PBA ; 
 
 and since this result is independent of the direction of AP, 
 the time down all chords drawn from A will be equal. 
 
 COE. Similarly, it may be 
 shewn that the times down all 
 chords terminating in 5 are equal. 
 
 The above result leads to the 
 solution of several curious pro- 
 blems of lines of quickest descent. 
 We give one such problem. 
 
 To find ike line of quickest 
 descent from a given point A to a given circle. 
 
 Construct a circle of which A shall be the highest point, 
 and which shall touch the given circle in some point P, then 
 will AP be the line of quickest descent required. For if 
 ^e join A with any other point of the given circle, the joining 
 line will be longer than the part of it intercepted by the second 
 circle, and therefore the time down the joining line would 
 be longer than the time down the corresponding chord of the 
 
216 OF UNIFORMLY ACCELERATED MOTION. 
 
 second circle, i. e. greater than the time of descent down 
 AP; consequently AP is the line of quickest descent horn 
 A to any point of the given circle. 
 
 If the circle be drawn it can easily be shewn that AP 
 produced will pass through the lowest point of the given 
 circle, — whence we have the simple rule : Join A with the 
 lowest point of the given circle, and the part of this line 
 without the circle is the line required. 
 
 81. An accurate knowledge of the numerical value of ^ the 
 accelerating force of gravity is of great importance, and various 
 methods have been employed to determine it. If a body were 
 sliding down a smooth inclined plane of elevation a, the ac- 
 celeration upon it would be g sin a ; so that by diminishing 
 a sufficiently, the force acting upon the body might be reduced 
 so as to admit of the inotion being observed, without the latv 
 of the motion being aifected. This method of determining g 
 was suggested by Gralileo, — but since no surface can be obtained 
 sufficiently smooth, the method does not practically admit of 
 great accuracy. 
 
 A machine invented by Atwood for the purpose of making 
 observations on the laws of falling bodies, leads to results 
 much more trustworthy than the preceding. 
 
 We will here give a short description of the machine. 
 
 82. Two equal weights P, Q are attached to the extre- 
 mities of a fine thread which passes round a pully C. The 
 axis of rests in a horizontal position on four wheels, of 
 which two only are represented in the figure; the object 
 of these wheels being to diminish the friction on the axis 
 of C, which they do very considerably, since the friction of 
 
ATWOOD S' MACHINE. 
 
 217 
 
 rolling is much less than that of ruVbing. Hence they are 
 cbH&A. friction wheels. 
 
 Now P and Q being equal would 
 be in equilibrium, but if a weight R 
 be placed upon P, it would begin 
 to descend subiect to an acceleration 
 
 R * 
 
 F+Q + M '^' ^®^® ^'^^' ^^^' ^^ *^® 
 motion of the puUy were neg- 
 lected. It is found that the rotation 
 of the heavy puUy G has the effect 
 of adding something to the weights 
 moved (viz. P + Q + E), without 
 altering the force which produces 
 motion, (viz. the weight of R). At- 
 wood determines by experiment what 
 this is, — call it W, — then the accele- 
 ration upon P becomes 
 
 . . P+Q + R+W ^^ (=-^) ™PP°''' ^"^ 
 by diminishing R sufficiently, this 
 may be reduced to as small a quantity 
 as we please. 
 
 AB is a vertical graduated bar, 
 and 8, T are two platforms capable 
 of motion backward and forward along 
 the bar, and of being fixed in any position by screws. The 
 platform 8 is pierced so that P can pass freely through it, 
 but not R. 
 
 If now the system be allowed to start from rest with P at 
 a given position (say A), P will move through the space AS 
 
218 OF UNIFORMLY ACCELERATED MOTION. 
 
 subject to the uniform acceleration yj — and B being caught off 
 at S, P will move on through 8T uniformly with the velocity- 
 acquired. The times occupied in moving through A 8 and BT 
 are observed with considerable accuracy by a contrivance of 
 clock-work attached to the machine. 
 
 83. The results of numerous experigaents made with 
 Atwood's machine, lead to the conclusions that gravity has 
 a uniform accelerating effect, and that its numerical value is 
 that stated in Art. 43. The most trustworthy results however 
 are (as there stated) to be obtained from experiments on 
 pendulums, but they are of too refined a character to be 
 discussed here. 
 
( 219 ) 
 
 CHAPTEK IV. 
 
 OP THE MOTION OF PROJECTILES. 
 
 84. In the present Chapter we shall consider : 
 
 (i) The projectile as a single heavy particle, (ii) that the 
 accelerating force of gravity is uniform, and acts in the same 
 direction at all points of the path of the projectile ; (iii) that 
 the effect of the rotation of the earth is neglected, and 
 (iv) that the motion takes place in vacuo — ^no account being 
 taken of the resistance of the air. See Art. 94, 
 
 85. A iody projected in any direction which is not vertical, 
 and acted on iy the force of gravity only, will describe a 
 parabola. 
 
 Let the body be projected from the point A in direction 
 A T with Telocity v. Draw AV y 
 vertical and downwards, and 
 let P be the position of the 
 body at any time t after the 
 instant of projection. 
 
 Let the motion of the body / 
 be referred to the directions A T, 
 ^F(Art. 19), and draw FT, 
 
 PV parallel to ^ F, A T; — now the motion being at any and 
 every instant referred to the directions AT, A V — the force of 
 gravity will have a uniform accelerating effect {g) in direction 
 ^Fand there will be no acceleration in direction AT, we 
 shall have therefore 
 
220 OF THE MOTION OF PROJECTILES, 
 
 FV=AT=vt (Art. 49), 
 Ar=^gt^ (Art. 68), 
 
 whence PV = vY =—.AV. 
 9 
 
 This relation between PV and A V shews that the path 
 AP is a parabola whose axis is vertical, and directrix con- 
 sequently horizontal; AY being a diameter, and AT the 
 
 tangent at A, the parameter at A being = — . 
 
 Cor. 1. If A be the space due to the velocity of pro- 
 jection V, (i. e. the space through which a body must fall freely 
 from rest under the action of gravity, in order to acquire the 
 velocity?;,) v'='2gh; wherefore P 7"^= 4A . -4 7". Hence 4A 
 is the parameter at A, and therefore h Is equal to the vertical 
 distance of A below the directrix. 
 
 Cor. 2. The result of Cor. 1 may be thus interpreted : 
 " The velocity of projection of a projectile is the same as 
 would be acquired by a body falling freely from the directrix 
 to the point of projection." 
 
 And further, since the body after passing through any 
 point of Its path will move in the same way as if it had been 
 projected from that point with the velocity it then has, and 
 in the direction In which it is then moving, — hence, "the 
 velocity of a projectile at any point P of its path is equal to 
 that due under the action of gravity to the vertical distance of 
 that point from the directrix." 
 
 Ohs. Let the horizontal plane which passes through 
 the point of projection A meet the parabola again in H, and 
 let T be the time of passing from A to M^ AH= R, — then 
 
OF THE MOTION OP PEOJECTILES. 
 
 221 
 
 B, T are called the range, and time offiight of the projectile on 
 the horizontal plane; and if a be the angle which the direction 
 of projection makes with the horizon, the angle a is called the 
 elevation of the projectile. 
 
 86. If the motion of the body be estimated vertically and 
 horizontally^along Ay and Ax, — the velocity of projection 
 vertically is wsina, and horizoii- 
 tallyis i)cos a; — the horizontal 
 velocity will remain uniformly 
 equal to v cos a during the 
 motion, since there is no force 
 in direction Ax; the vertical 
 velocity will gradually be re- 
 duced to zero by the action of 
 gravity, and the body is, then 
 
 at its greatest height z above the horizontal plane AH, but 
 the continued action of gravity will generate velocity down- 
 wards, and bring the body to the plane at H after a time 
 equal to that in which it moved from A to the highest point. 
 We shall have the following results, 
 
 if T be the time of moving from A to the highest point 
 
 = time in which the initial vertical velocity v sin a, is destroyed 
 by force of gravity g, 
 
 V sma 
 
 and z = 
 
 (v sin a)^ 
 
 9 
 Hence T= 
 
 '^9 
 2v sin a 
 
 9 
 
 .(I), 
 •(ii), 
 
 and the horizontal velocity is uniform and equal to v cos a ; 
 2^" sin a cos a «''sin2a 
 
 E= T,vcoaoL = - 
 
 9 
 
 (lii). 
 
222 
 
 OF THE MOTION OP PEOJECTILES. 
 
 TT 
 
 This result is the same if — — a be put for a ; shewing 
 
 that there are two directions in which a body may be pro- 
 jected with a given velocity, so as to have the same horizontal 
 range. 
 
 For a given velocity of projection v^ the horizontal range 
 R will be greatest when sin 2a = 1 ; i. e. when a = 45°. 
 
 Again, the latus rectum of the parabola is ^^ parameter at 
 the highest point, and the velocity at the highest point being 
 = V cos a, the distance of that point from the directrix is 
 
 V cos a 
 
 ^9 
 
 (Cor. 2), 
 
 2*^ cos cc 
 hence the latus rectum = = 4A cos^ a. 
 
 87. To find the range {R) and time of flight {T) of a pro- 
 jectile on an inclined plane. 
 
 Let i be the inclination of the 
 plane to the horizon, 
 
 a the elevation of projection ; 
 then initial velocity perpendicular 
 to the plane AP= v sin (a — i), 
 initial velocity parallel 
 
 to the plane AP= v cos (a — i) ; 
 
 accelerating force of gravity per- 
 pendicular to the plane = g cos i, 
 accelerating force of gravity parallel ^ 
 to the plane = g sin i; 
 and these two resolved parts of gravity are constant. 
 
 Hence if T be twice the time in which the velocity 
 
RANGE AND TIME OP FLiaHT. 223 
 
 V sin (a — i) would be generated or destroyed by the force 
 g cos*; we bave 
 
 y_ 2»sin(a-t) _ 
 
 ^ cos V V'' 
 
 .'.R = v cos (a - 1) . r- ^g sin * . T" (Art. 70), 
 
 2d'' cos a sin (a — * ) 
 
 = T~- '~~ (ii), 
 
 ^cos * ^ '' 
 
 by substituting for T, and reducing; 
 
 or we might obtain B, thus 
 
 7?— AP — '^■^ — •" "^os g . T 
 cos* cos* 
 
 (since the horizontal Telocity v cos a is uniform) 
 
 _ ^v" cos a sin (a — * ) 
 
 .(iii). 
 
 g cos * 
 
 COE. The greatest perpendicular distance of the particle 
 from the plane will be when the velocity is entirely parallel 
 to the plane, 
 
 i. e. after a time —■ — ^ — i—^; 
 g cos * 
 
 and this perpendicular distance = — - — ^ — ^ . 
 ^ ^ 2^ cos* 
 
 Further, by putting (iii) in the form 
 
 ■n _v^ {sin (2a — *") — sin*'} 
 g COS * 
 
 we see that for a given velocity of projection the range is 
 greatest when sin (2a — *') = 1, 
 
 IT TT * 
 
 2'°^"" = 4+2' 
 
 i. e. 2a — *' = — , or a = - + 
 
 in other words, when the direction of projection bisects the 
 angle between the plane and the vertical. 
 
224 
 
 OF THE MOTION OP PEOJECTILKS. 
 
 88. To find the equation to the path of a projectile referred 
 to horizontal and vertical co-ordinate axes. 
 
 Let A be the point of pro- 
 jection, Ay vertical, and Ax 
 horizontal in the plane in which 
 the projectile moves. 
 
 AN=x, NP = y, the hori- 
 zontal and vertical co-ordinates 
 of the particle at time t after 
 projection, v the velocity and a 
 the elevation of projection. Let NP he produced to meet in 
 rthe line ^7 which represents the direction of projection. 
 
 Then the horizontal velocity = d cos a, which remains 
 uniform, and the initial vertical velocity = u sin a. 
 
 And we have 
 
 x = AN=vcosa..t (i). 
 
 And NP represents the space passed over parallel to Ay 
 in time t by a particle projected with a velocity v sin a, and 
 retarded by a force g — hence by Art. (70) 
 
 y = NP = vsva.a..t-\ge (ii). 
 
 Eliminating t between (i), (ii), we get 
 
 y = x tan a. — 
 
 2?;'' cos'' a' 
 
 (iii), 
 
 which is the equation to the path required, and represents 
 a parabola. 
 
 If h be the height due to the velocity of projection t;" = 2gh, 
 and equation (iii) may be written 
 
 y = x tan a — —; ~- . 
 
 " Moos'' a 
 
 (iv). 
 
EEFEERED TO CO-OEDINATE AXES. 225 
 
 From equation (iii) or (iv) the elements of the paraholic 
 path may e^ily be deduced. 
 
 Cob. 1. The equation « = a tan a — -^ n- may by a 
 
 ^ " 4Acos''a •' •' 
 
 little reduction be put in the form 
 
 {x - 2A sin a cos a)^ = 4A cos" a {h sin** a - y), 
 
 from which we may readily infer that the co-ordinates («„, y^ 
 of the vertex of the parabola are 
 
 x^ = ih sin a cos a, y^ = A sin^ a ; 
 
 and the latus rectum = 4^ cos' a. 
 
 Cob. 2. If we make y = in equation (iv), we get 
 
 G = a; tan a — tt — -5-; 
 4a cos a 
 
 i.e. 03 = 0, or a; = 4A sin a cos a ; 
 
 the former value of x indicates the point of projection, the 
 latter gives the range on the horizontal plane Ax, and accords 
 with the result obtained in Art. (8B). 
 
 Cob.. 3. If ^ be the angle which the direction of motion 
 of the projectile at a time t after projection makes with the 
 horizon, its altitude above the point of projection being y and 
 its velocity v — we shall have 
 
 vertical velocity = «' sin <j^ = « sin a—gt, 
 
 horizontal velocity = v' cos <^ = u cos a, 
 
 whence we get 
 
 . t) sin a — oi 
 
 tan ffl = '— , 
 
 ^ V cos a 
 
 P. M. 15 
 
326 OF THE MOTION OF PROJECTILES. 
 
 v'^ = {v sin a —gtY_ + [v cos a)" 
 = «' — 2y (w sin a . « — \gf) 
 = v^ — Igy. 
 
 r 
 
 89. Motion of a projectile on a smooth fixed inclined plane 
 under the action of gravity. 
 
 With the diagram of Art. 88, let Ax, Ay be rectangular 
 axes on the inclined plane (elevation =i), Ax being drawn 
 horizontal and Ay up the plane — Ay will be a line oi greatest 
 slope on the inclined plane. 
 
 If the projectile start from A with a velocity v in direction 
 ^r along the plane {TAx = a) the acceleration of P will be 
 zero parallel to Ax, — gsmi parallel to Ay — and gcoai per- 
 pendicular to the plane, the equation to the path will be as in 
 Art. 88, 
 
 a sin i . a;" 
 
 «= a; tana — ^-2 =— , 
 
 ^ 2v cos' a ' 
 
 a parabola, the elements, of which can be obtained as in the 
 previous article. 
 
 If v' be the velocity at any point P {x, y) of the path, 
 j> the angle which the tangent at P makes with Ax, t the time 
 of motion from A to P, z the vertical eXiitvAt of P above A, 
 so that s = y sin i, we shall have 
 
 d' cos ^ = «cosa (i), 
 
 v'sin (^ = «sina— ^sin^. « (ii), 
 
 y = vs,\D.a. .t~\gim.i. f (iii), 
 
 a! = j>cosa.< (iv); 
 
PEOBLEMS. 227 
 
 •from (i), (ii), (ill), -we get 
 
 v'^ — i? — 1g sin i .y= v' — '2gz, 
 
 a result whicli shews that the change of velocity is the same 
 as if the projectile had moved freely under gravity, through 
 the same vertical height. 
 
 90. Peob. a body is projected with a given velocity v 
 from a given point, to find the direction of projection that it may 
 strike another given point. 
 
 Employing the notation of Art. (88), 
 
 Let A be the point of projection, P the point through 
 •which the body is to pass, h the height due to the velocity of 
 projection, and a the required elevation of projection. 
 
 Then the equation of the path is 
 
 a? 
 
 y — x tan a — -; w- : 
 
 ^ 4Acosa 
 
 if (a, 5) be the co-ordinates of P, we have the equation 
 6 = a tan a — 
 
 AJi cos" a ' 
 
 ox h = a tan "^ — tt (1 + tan" a), 
 
 from which to determine a. 
 
 This equation is a quadratic in tan «: — ^when the two roots 
 are real and unequal, there are two directions of projection 
 which will satisfy the problem ; — ^when the two roots are real 
 and equal, these two directions coincide, — and when the roots 
 are unreal the problem is impossible, i.e. there is no direction 
 in which the body could be projected with the proposed velo- 
 city so as to pass through the given point. 
 
 15—2 
 
228 OP THE MOTION OP PROJECTILES. 
 
 This problem admits of a simple geometrical constniction. 
 From the point of projection A 
 draw J.5^Tertical and = h, throtigh 
 H draw SL horizontal, then ML 
 will be the directrix of the para- 
 bolic path Art. (85), Cor. The 
 problem then resolves itself into 
 this^-to construct a parabola which 
 shall pass through each of the points 
 A, P and shall have HL for its directrix. With A and P as 
 centres describe circles touching the line HL, and let 8 be 
 one of the points in which these circles intersect. 
 
 Then since 8A = AII and 8P= PL, A, P are points in 
 a parabola whose focus is 8 and HL the directrix, and ii AT 
 bisect the angle HA8 it. is. a tangent to the parabola at A, 
 and consequently indicates the direction of projection. 
 
 If 8' be the other point in which the circles intersect, and 
 AT' bisect the angle HA8, then AT' indicates another 
 direction of projection which will equally satisfy the problem. 
 If the circles touch each other, then 8, 8' coincide, and there 
 is but one parabola and one direction of projection. If the 
 circles do not meet there exists no direction of projection 
 which will satisfy the problem. 
 
 The student will have little diflSculty in reconciling the 
 results of the above analytical and geometrical solutions of 
 this problem. 
 
 COE. The locus of points P to any one of which there 
 is but om parabolic path for the particle projected from A 
 with given velocity, is a parabola having A for its focus, and 
 H its vertex. 
 
PROBLEMS. 229 . 
 
 91. We have seen in Art. (89) that when a particle 
 moves on a smooth inclined plane the change of velocity., 
 in passing from one position to another is the same as if the 
 particle moved _^ee?y under the action of gravity through the 
 same vertical space. We shall see in the next chapter that 
 the same conclusion is true, if the particle be moving on a 
 smooth curve. 
 
 If then a particle moving 6n a smooth plane or curve 
 quit it and subsequently describe a parabolic trajectory un- 
 der the action of gravity — and if v = >J^gh be the velocity 
 at any point A of the path on the surface, h will be the 
 vertical altitude above A of the directrix of the parabola : — so 
 that we may find the position of this directrix without neces- 
 sarily determining where the particle quits the surface. 
 
 We will make use of the above result in the following' 
 problem. 
 
 92. An inclined plains infixed on a table, and from the. 
 
 foot of it a hody is projected upwards along the plane with the 
 
 velocity due to the height h ; after passing over the top of the 
 
 plane the hody strikes the table at distance z from the foot of 
 
 the plane ; — shew that if the length of the plane be I, and a 
 
 7r 
 its inclination to the horizon be <—, the greatest value of z for 
 
 qiven values of h and a. is —. , and corresponds to the 
 
 ^ •' sm a cos a. -^ 
 
 value l = 2h . 
 
 cos a. 
 
 Let AB (= l) be the inclined plp,ne, AP the table. Draw 
 AD = h vertical, ZT horizontal, produce AB to meet DT'm T 
 and draw 2!Pat right angles to J.?* meeting the table in P» 
 
230 
 
 OF THE MOTION OP PEOJECTILES. 
 
 Now the particle projected from A with velocity = >j2gh 
 after quitting the plane at 5 will describe a parabola to which 
 £T is a tangent and of which J)T is the directrix. 
 
 x>i 
 
 
 ■-■^^! 
 
 iLF 
 
 \ 
 
 l\ 
 
 Also since tangents to a parabola which meet in the direc- 
 trix are at right angles to one another, therefore TF touches 
 the parabola somewhere : — since then the body cannot pass 
 beyond the line TP, the range on the plane AP will evidently 
 be greatest when it touches at P, and we have 
 
 z = AP=AT.seca = 
 
 h 
 
 2h 
 
 sin a cos a sin 2a * 
 
 Also BP will pass through the focus S, T8 will be perpen- 
 dicular to BP, and the angles which TP makes with PB and, 
 the vertical are each = a, 
 
 whence 
 
 I 
 
 z 
 
 .1 = 
 
 AB 
 'AP' 
 
 cos 2a 
 cos a ' 
 
 2h cot 2a 
 cos a 
 
 Further, BP would not meet the table to the right of A 
 
REMARKS. 231 
 
 if 2a be > - , hence in order that the problem may be posi- 
 
 2 
 
 sible a must be < -r • 
 4 
 
 Senate-Souse Prob. Jan. 18, 1854. 
 
 93. Prob. A particle {whose elasticity is e) is projected 
 with velocity v at an elevation a from a point in a horizontal 
 plane ; to find the time in which the vertical velocity will be 
 destroyed hy successive rebounds and the total horizontal range 
 described in that time. 
 
 The particle will describe a series of parabolic curves in 
 one plane ; the initial vertical velocity being v sin a, and the 
 vertical velocities at the successive rebounds being ev sin a, 
 ^v sin a, e'u sin a, &c. 
 
 Now the time of describing any one of these parabolic 
 
 curves in which the initial vertical velocity is m, is = — , 
 
 Art. (86). Hence the whole time which elapses before the 
 initial vertical velocity v sin a is destroyed by successive 
 rebounds is 
 
 And since the horizontal velocity continues uniform and 
 = V cos a, the whole horizontal range described in this time 
 will be 
 
 2v sin a u° sin 2a 
 
 . ?; cos a != • 
 
 g{l-e)— g{l-ey 
 
 The particle will afterwards move along the horizontal plane 
 (supposed smooth) with the uniform velocity 'v cos a. _ 
 
232 OF THE MOTION OP PROJECTILES. 
 
 COE. The vertical velocity at the beginning of the m"" 
 curve will be = e"~'. v sin a, and if a„ be the elevation at that 
 time, we shall have, since v cos a is the horizontal velocity, 
 
 vertical velocity , , ^ 
 
 tan a„ = ^j — -. —j — , — rr = e • tan a. 
 
 honzontal velocity 
 
 Also the time of describing the first n parabolas 
 
 9 9 l-« 
 
 and the sum of the ranges of these n parabolas 
 
 2wsina l-e" v'sInSa 1-^ 
 
 = . « cos a = . :; ; 
 
 a l-e a l-e ' 
 
 94. The theory of the motion of projectiles given in this 
 Chapter depends upon the suppositions stated in Art. 84, 
 which are all inaccurate. The force of gravity without the 
 Earth's surface varies inversely as the square of the distance 
 from the centre of the Earth ; but the height to which a body 
 can be projected from the surface is so small, that the varia-. 
 tion of the force arising from the change of the distance from 
 the centre may be safely neglected. The direction of the 
 force is everywhere perpendicular to the horizon, — but if per- 
 pendiculars were drawn to the horizon at points on the Earth's 
 surface five miles apart, the angle between them would not 
 exceed 1', so that any error arising from the non-parallelism 
 of the force of gravity may be neglected ; and the same may 
 be said of the very small errors arising from the rotation 
 of the Earth about her axis, and her motion of translation 
 in space about the Sun. The principal cause of error is the 
 
EEMAEKS. 233 
 
 resistance of the air, and this is so considerable as to render 
 the conclusions drawn from the theory almost entirely inap- 
 plicable in practice. From experiments made to determine 
 the motion of cannon-balls, it appears that when the initial 
 Telocity is considerable, the resistance of the air is 20 or 
 30 times as great as the weight of the ball ; and the horizontal 
 range is often a small fraction of that which the preceding 
 theory gives. Such experiments have been made with great 
 care, and shew how little the parabolic theory is to be de- 
 pended upon in determining the motions of military projectiles. 
 
 From a long series of experiments made at Woolwich, 
 Dr Hutton arrived at the conclusion that the velocity v of a 
 cannon-ball on quitting the gun could be nearly expressed by 
 
 /op 
 
 the formula v = 1600 y' _ , Pbeing the weight of the charge 
 of powder and IF that of the ball. 
 
 And further, if the projectile be of finite size, and have a 
 rotatory as well as &, progressive motion, the resistance of the 
 air, which acts along the surface of the body (or tangentially), 
 will in general change its direction, or the plane of its motion, 
 or both. For this resistance increases with the velocity and 
 the density of the air, and will consequently be greater on 
 that side of the body where the rotatory and progressive 
 motions conspire, than on the other side where they oppose 
 each other : and the density of the air immediately in front 
 of the body is greater than behind it. 
 
 Another cause of irregularity will also exist if the ball 
 be not homogeneous — as for example if it contain air-bubbles 
 within, from imperfection in the casting — so that its centre of 
 gravity does not coincide with its centre of figure. 
 
234 OF THE MOTION OF PEOJEOTILES. 
 
 The non-symmetrical action of tlieSe causes on the body 
 will make it deviate from its plane of motion, except in the 
 single case when the axis of rotation coincides with the 
 direction oi progressive motion. On this principle has been 
 explained the irregular motion of a tennis-ball and the devia- 
 tion of a bullet from the vertical plane. It is in a great 
 measure remedied in the case of a rifle ball, since the rifling 
 of the barrel communicates to the ball a rotation about an 
 axis in the direction in which the ball is projected. 
 
 (See Eobins' Gunnery ; Hutton's Tracts ; Art. Gunnery 
 in the Encyclopmdia Britannica.) 
 
( 235 ) 
 
 CHAPTEE V. 
 
 MOTION ON A CUfiYE. 
 
 95. When a body moves along a smooth curve the 
 curve exerts a pressure or reaction upon the body at every 
 point, but since this reaction is always perpendicular to the 
 curve, it has no tendency to accelerate or retard the body. 
 In order to determine the velocity of the body in any position 
 we must resolve the forces upon the body in direction of the 
 motion at successive instants, and examine the effect of these 
 resolved forces. 
 
 96, An melastio particle descends down a smooth curve in 
 a vertical plane under the action of gravity, to find the velocity 
 of the particle in any position. 
 
 We may regard the curve as the limit of a polygon whose 
 sides are equally inclined to one 
 another, by supposing the number of 
 sides to be indefinitely increased, 
 and the angle between consecutive 
 ones to become evanescent. 
 
 Let AA^...A„ be such a poly- 
 gon; draw A^a^, A^a^, ^jflg... per- 
 pendiculars on the vertical line 
 through A. 
 
 Let 6 be the angle between suc- 
 cessive sides of the polygon which 
 are not necessarily of equal length, 
 
236 
 
 DESCENT ON A CUEVE. 
 
 u the velocity at A in direction AA^, 
 
 
 A. 
 
 AA„ 
 
 Vn A„ 
 
 Then we shall have (Art. 89) 
 v^' = u^ + 2ff. Aa^ 
 similarly, v^ = v^ cos^ ^ 4- 2^' . a^,.^ 
 
 •' A^.A„. 
 
 whea the particle 
 comes to A^ it is de- 
 flected in direction 
 A^A^ and starts along 
 A^A^ with velocity 
 
 v„^ = «Vi cos" 6+2ff. a^jtz^ J t)j cos 
 
 adding and transposing we get 
 
 Vn' + « + «/ + - + ^Vi) sin= e = u' + 2ff.Aa, (i). 
 
 Now if a. he the angle hetween the directions of motion at 
 A and A„, and v the greatest of the velocities v^, v^.., 
 
 and Aa„ = h, we have a = (n — l)d, 
 
 and {v,^ + «/ + ...+ t;V.) sin" e<{n-l)v" sin" 61 ; 
 
 ,, fsmff 
 
 :. <o.ev" 
 
 ■\~T 
 
 and this vanishes in the limit when n is indefinitely increased, 
 a remaining finite, (in which case the polygon becomes the 
 curve) ; 
 
 .-. h fortiori (v^" + v^^ + ...+ ^VJ sin" 
 
 vanishes in the limit, in comparison with v„". 
 
 Hence in the limit when the polygon becomes the curve, 
 the equation (I) becomes 
 
 v^ = u'+2gh, 
 
MOTION ON A CURVE. 237 
 
 wKich expresses the velocity at any point ofi the curve in 
 terms of the initial velocity, and the vertical height through 
 •which the particle has fallen ; or suppressing the suffix, 
 
 97. Ohs. In the above investigation we have supposed 
 the particle inelastic and moving on the concave side of the 
 curve, towards which the force of gravity pulls the particle; 
 these suppositions being made in order that, the particle may 
 remain in contact with the curve. We shall see hereafter that 
 a particle moving on a curve will, under certain conditions, 
 quit the curve; but the necessity of the supposition here 
 referred to would be obviated by supposing the polygon 
 AA^A^... to be a polygonal tube (becoming a curvilinear 
 one in the limit) of small bore,, just sufficient to allow the free 
 passage of the particle. The result arrived at for the velocity 
 at any point would hold good in this case, and will be equally 
 true for a particle moving either on the concave or convex side 
 of a curve, so long as ti remains in contact with the curve. 
 
 CoK. 1. If the particle start from rest at A, then m = 0, 
 &ndiv^ = 2gh; i.e. the velocity, acquired from rest, down a 
 smooth curve is equal to that which would be acquired by a 
 body falling freely through the same vertical height. More 
 generally- we may interpret the equation v"' = u^ + 2gh thus : 
 the square of the velocity at any point A„, is equal to the square 
 of the velocity at any other point A, increased ly the square 
 of the velocity which the force of gravity would generate in the 
 hody in drawing it from rest through the same vertical space. 
 
 This result it will be observed is independent of any par- 
 ticular form of the curve. 
 
 CoE. 2. If a body be projected up a curve, the vertical 
 
238 
 
 MOTION IN A CIRCLE. 
 
 height to which it will rise is equal to that through which it 
 must fall in order to acquire the velocity of projection ; for 
 the body in its ascent will be retarded by the same degrees 
 that it would be accelerated in its descent. 
 
 If u be the velocity at any point of a particle moving up 
 a curve, v its velocity after describing a vertical height h, we 
 shall have v' = u'— 2gh. 
 
 Hence if BAB" be a curve in a ver- 
 tical plane, the lowest point of which 
 is A, and the parts AB, AB' are similar 
 and equal, a body in falling down BA 
 will acquire a velocity which will carry 
 it up to 5* ; and the velocities at all equal 
 altitudes in the ascent and descent being 
 equal, the whole time of ascent will be 
 equal to the whole time of descent. 
 
 It is moreover obvious that when 
 the particle has arrived at B it will descend again to A and 
 rise to B, and so on continually; i.e. the motion will be a reci- 
 procating or oscillatory one, and the time of passing from B 
 to B' through the lowest point A is called the time of oscil- 
 lation. 
 
 Cor. 3. Let BAB be a circle (radius a) of which A is 
 the lowest point, ^0 the vertical radius, and ^iV' drawn per- 
 pendicular to J.0; w the velocity acquired by a particle in 
 descending from rest at B to the lowest point A; then we 
 shall have 
 
MOTION ON A CURVE, 239 
 
 i.e. the velocity at the lowest point varies as the chord of the arc 
 of descent. 
 
 The result will be the same if instead of the curve BAB 
 we suppose the particle attached to an inextensible string of 
 length OA, and fixed at 0. 
 
 98. Ohs. The time in which a particle will fall from 
 rest from a point B to the lowest point A will not, in most 
 cases, be the same for difierent positions of B. But if the 
 curve be a cycloid the time of falling to the lowest point will 
 be the same, whatever be the point from which the body- 
 starts; in other words, the time of oscillation in a cycloid 
 (whose axis is vertical and vertex downwards) is the same 
 whatever be the arc of oscillation. For this reason the cycloid 
 is called an isochronous curve. 
 
 This property of the cycloid is of great importance in the 
 theory and construction of pendulums. 
 
 We proceed to give a proof of it : but for the convenience 
 of the student we will first give a proof (in the following three 
 articles) of the properties of the cycloid which it will be neces- 
 sary for him to be acquainted with. 
 
 99, Bef. If a circle as TPS roll in one plane upon a straight 
 line GBD, any point P fixed on the circle -will trace out a curve 
 GPAD called a cycloid. 
 
 Let CPAD be the complete curve formed in one revolution of 
 the circle, G, D the points where the tracing point P quits and 
 returns to the line CD, SPT the position of the circle when the 
 tracing point is at P, BQA its position when the tracing point 
 P is furthest from the liae GD ; then it is evident that the parts of 
 the curve AG, AD will be equal and similar, — AB which bisects 
 GD at right angles is called the axis, GD the hose, and -4 the vertex 
 of the cycloid. 
 
240 PEOPERTIES OP THE CYCLOID. 
 
 Draw PQR perpendicular to AB, join PS, FT. 
 D B S 
 
 Since P starts from C, and every point of the arc SP has been 
 in contact with the line C'S, — the arc SP = SG ; and since the line 
 CB is equal to the semicircle BQA, which is = semicircle SPT, 
 therefore arc PT=BS = PQ, since BQ is parallel and equal to PS. 
 
 Hence if we suppose the circle to begin rolling from the posi- 
 tion BQA with the tracing point at A, when it arrives at any 
 position P, the axe AQ = BS=PQ. 
 
 (i) PTis a tangent to the cycloid at P. 
 
 Tor when the tracing point is at P, the generating circle is in 
 contact with the line CD at S, and this point S of the circle is at 
 rest for an instant, or the circle is turning about S, — consequently 
 P is moving perpendicularly to SP, i. e. PS is a normal to the 
 cycloid at P, and PT (which is at right angles to SP) is the 
 tangent at P. 
 
 (ii) The length of any arc AP starting from the vertex is 
 twice the chord AQ of the circular arc AQ cut off by the ordinate 
 PQF. 
 
 Let P,QiN, be an ordinate very near to PQF; draw ^F, YQt 
 tangents to the circle 2A. A,Q; then ^F is parallel to the base 
 (and also to P, §,iV_)— let YQ, AQ produced meet P,Q,F^ in t„q; 
 and draw tn perpendicular to Qq. 
 
 Kow A7, VQ being tangents to the same circle jl YAQ=l YQA, 
 and ^ YQA = opposite ^tQq, and /itqQ — alternate z. YAQ. 
 
MOTION ON A CURVE. 
 
 241 
 
 Hence ttQq = i tqQ, and .: tq=--tQ; consequently Qq = 2. Qn. 
 
 Further, the smaller QQ^ is taken, the more nearly does the 
 arc QQ^ coincide with its tangent Qt, and is ultimately equal to it 
 (Newton, Lemma vii). Hence tn is ultimately a small arc of a 
 circle, whose centre is A and radius A Q^ — i. e. AQ, — and Qn ulti- 
 mately measures the increment of the chord AQ. 
 
 Also Qq is parallel to the tangent to the cycloid at P, and is 
 therefore ultimately equal to the arc PF^. 
 
 Hence the increment PP, of the arc of the cycloid is ultimately 
 twice the corresponding increment of the chord A Q, — and the arc 
 AP and chord j4§ begin together at A, — therefore arc AP= 2 . chord 
 AQ = 2 chord TP. 
 
 Cor. 1. Since AQ' = AN.AB, .: AP = 2J {AB . AN). 
 
 CoK. 2. The (M-c AG = 2 . AB. 
 
 100. (iii) To make a pendiClv/m osdllate in a given cycloid. 
 
 Let AB be the axis and 
 
 DC the base of the given 1 B_ T 
 
 cycloid, and let EG be a 
 semicycloid exactly equal to 
 DA placed with its vertex 
 at G and base EF parallel 
 to BG ; — ED another semi- 
 cycloid equal to GA placed 
 with its vertex at D and 
 base parallel to DB. 
 
 Let ^^iS', SPT be any positions of the generating circles of the 
 cvdoids touching each other at S; Q, P the positions of the tracing 
 points, — join QS, SP. 
 
 Then arc PS = SG = BF = axe SQ ; hence since the circles are 
 equal, the angles PSG, QSB are equal, and therefore PSQ is a 
 straight line. 
 
 P.M. 
 
 16 
 
242 MOTION ON A CUEVE. 
 
 But QS is a tangent at Q to the cycloid SQC, and FS is normal 
 at P to the cycloid CFA. 
 
 Also the arc GQ = 2. chord SQ = PQ. 
 
 Hence if we suppose a string of length = length of semicycloid 
 EQG to be fastened at E and applied to the cycloid EQG, and 
 if it be then unwrapped, being kept always stretched, it will 
 always be a tangent to the cycloid EQG, and its extremity will 
 trace out the cycloid GA. 
 
 "We have then this practical way of making a pendulum 
 ■vibrate in a cycloid. 
 
 Let two equal material semicycloids EQG, ED, be placed so 
 as to have a common vertical tangent at E, and let a fine string 
 of length equal to the semicycloid EQG be fastened at E, and 
 have a heavy particle attached to its other end P. The particle 
 will oscillate in the cycloid GAD, the string unwrapping from 
 EG as P describes CA and then wrapping itself on ED, whilst P 
 describes AD, — and vice versd, continually. 
 
 101. (iv) The radius of curvature at any point P of the 
 cycloid = PQ = 2 . PS = 2 . normal ; as is evident from the pre- 
 ceding article, or we may arrive at this result independently thus 
 with the fig. Art. 99. 
 
 Join £Q', £Q~-the latter cutting AQ' in o, let PS, P'S' the 
 normals at P, P' intersect in 0' the centre of curvature at P, 
 then since PO', P'O' are parallel to PQ, BQ' respectively, and 
 PP'=2.^o ultimately, 
 
 .: P0' = 2.Bo = 2.BQ = 2.PS ultimately, 
 i. e. rad. curv. = 2 . normal, 
 
 102, To find the time in which a particle will fall down 
 any arc of an inverted cycloid. 
 
MOTION ON A CYCLOID. 
 
 243 
 
 Let V be the point from which the particle starts from 
 rest ; VS horizontal meeting the axis of the cycloid AB in E. 
 
 On AH describe a circle, aiid let the ordinates PN, P'N' of 
 two contiguous points meet this circle in j, g'; join Sq, Hq', 
 and Aq cutting Hq in h. 
 
 Now 
 
 arc 
 
 AP=2^iAB.AN)=2^{AB.^) = 2Aq.^[^, 
 similarly arc AF = 2Aq' aJ (jj^ > 
 
 ,.PP' = 2[Ag:-A^)^i^. 
 
 Again, since the particle starts from rest at F, the velocity 
 at P= velocity acquired in falling freely through the vertical 
 height HN 
 
 and since PP' is very small the velocity of the particle whilst 
 describing PP' will be very nearly uniform and equal to its 
 velocity at P, and the smaller PP' is taken the more nearly 
 
 16—2 
 
244 MOTION ON A CYCLOID. 
 
 ■will this supposition be true ; also, on the same supposition 
 Aq — Aq may be ultimately taken to be = bq, — since Aq'b 
 being = 90°, Ah = Aq' ultimately, and .'. Aq — Aq = hq. 
 
 arc PP' 
 Hence time of describing PP'= — j — r- — — =, ultimately, 
 
 -(^.-^.V(S)-^V(:i 
 
 ah) 
 
 since -j^ = circular measure of ^ qSq, ultimately) ; 
 
 i.e. the time of describing a-nj small arc PP' varies as the 
 circular measure of the corresponding angle qHc^. 
 
 If then we take the sum of successive small intervals 
 starting from F, we get the time of describing VP, — and the 
 sum of the corresponding small angles is = ^ V^i, 
 
 whence time of describing VP= /. VHq .a /( ) . 
 
 CoE. 1. When P comes to ^ the '■ VHq becomes VHA = 
 
 IT 
 
 2' 
 
 .•. time from Fto ^ = — . / ( ) . 
 
 The body after coming to A will ascend the opposite 
 semicycloid AD to a point V such 'that AV = A V; and the 
 time of ascending A V will be equal to the time of descending 
 VA. Hence the tim,e of a complete oscillation from Fto V is 
 
 -V -^ 
 
 ff /' 
 
TIME OF OSCILLATION. 245 
 
 COE. 2. Since the time of oscillation in a cycloid does not 
 depend upon tlie particular point from which the body starts 
 from rest, the time is the same whatever be the arc of oscilla- 
 tion, — in other words, the curve is isochronous. 
 
 CoE. 3. If two equal semicycloids EC, ED (fig. Art. 100) 
 be placed in contact at E with their common tangent vertical, 
 and a string of length equal to either of them be fastened at 
 E, and have a heavy particle attached to the other end, — this 
 particle will oscillate in the cycloid CAD in exactly the same 
 way as a free particle moving on a material cycloid CAD. 
 
 If I be the length of the string, i.e. of the pendulum, 
 1 = AE=2AB; and the time of an oscillation from rest to 
 rest will be 
 
 = ,rA/-. 
 
 9 
 
 Hence at the same place on the Earth's surface the time of 
 oscillation = \J {length of the pendulum). 
 
 CoE. 4. The cycloid for a short distance from A will very 
 nearly coincide with its circle of curvature at A, which is the 
 circle whose centre is E and radius AE. 
 
 If then a pendulum of length I oscillate in a circular arc 
 
 of very small amplitude, the time of oscillation = tt a/ — . 
 
 CoE. 5. If I be the length of the seconds pendulum, 
 i.e. of the pendulum which oscillates from rest to rest in 
 a second, 
 
 T the length of the pendulum which oscillates once in n seconds, 
 — we have . _ _ 
 
 '' .: l'=n\l. 
 
246 OSCILLATION OP 
 
 103. The length of the seconds pendulum in the latitude 
 of London has been found by experiment to be 39'1386 
 inches : — from this value of I we can find g the accelerating 
 
 force of gravity, for we have 1 = tt a/ — ; 
 
 .-. ^ = 7^^ Z = 386-28 inches = 32*19 feet. 
 
 COE. If ff, g be the force of gravity at two places A, B 
 where the same pendulum beats n, n! times respectively in the 
 same given time, we can easily compare g, g' in terms of n, n. 
 
 For if T be the given interval, we have 
 
 n 
 
 ■-Vj. 
 
 T 
 
 -'^\ 
 
 1 1 
 
 
 - ©'= 
 
 J. 
 '9' 
 
 
 
 -g_n'' 
 
 1 = 
 
 n'- 
 
 n^ 
 
 n —n n +n ^ n —n , 
 
 = . = 2 nearly, 
 
 n n n 
 
 ii n ~ n be small compared with n. 
 
 104. A seconds pendulum is taken to the top of a mountain 
 ofheigTit h ; to find the number of heats it will lose in a day. 
 
 Suppose the force of gravity to vary inversely as the 
 square of the distance from the centre of the Earth. 
 
 Let r be the Earth's radius, h = the height of the mountain, 
 g, g' the force of gravity at the foot and top of the mountain. 
 
A PENDULUM. 
 
 247 
 
 if t, i be the time of an oscillation at the foot and top of the 
 mountain, 
 
 t = ir 
 
 I . I I 
 
 and if n, n' be the number of beats in the same time at the 
 foot and top respectively, 
 
 nt = n't' ; 
 n_t' 
 ' ' n t 
 
 g r r 
 
 n— n 
 
 = 1- 
 
 1 h 
 
 —^ = - nearly; 
 
 if A be 1 mile, r = 4000 ; n = 24.60.60, 
 24.60.60 
 
 n—n 
 
 4000 
 
 -=21-6; 
 
 that is, a seconds pendulum would in this case lose about 
 21*6 beats in 24 hours. 
 
 N. B. For points outside the Earth, the force of gravity varies inversely as 
 the square of the distance from the centre of the Earth : — for points within the 
 Earth the force of gravity varies as the distance from the centre, 
 
 105. When a particle moves on a plane curve under the 
 action of any force, to find an expression for the acceleration at 
 any point of its path in the direction of the normal. 
 
 Let V be the velocity of the par- 
 ticle at any point P of its path, PO the 
 normal, PT the tangent at P. Take 
 PQ any small arc described in time t, 
 and draw Q8 perpendicular to PT, and 
 therefore parallel to PO. As the par- 
 ticle moves from P to ^ the velocity 
 and acceleration will in general vary. 
 
248 MOTION ON A CURVE. 
 
 Let v, v" be the greatest and least velocity estimated 
 parallel to FT, as the particle moves from P to ^ ; f, f" the 
 greatest and least acceleration estimated parallel to PO ; then 
 Q8 being the sp'ace through which the particle is drawn in 
 direction PO in the time < by a force always intermediate 
 to /', /", we shall have 
 
 Q8>U"^<:\f'f, 
 
 and PS>v"t<v't; 
 
 therefore ^-^>^<^. 
 Now when the arc PQ is taken continually smaller and 
 
 if2 '2 a - 
 
 smaller, each of the expressions -^ , -^, becomes -^ in the 
 
 poa 
 limit, and - — -j-^ in the same limit becomes the radius of 
 
 curvature at P(=/3 suppose). (See Evans's Newton, p. 15.) 
 
 Hence -^ = />, or /= — , the expression for the normal 
 acceleration' required. 
 
 106. By the first law of motion we know that if the 
 force acting upon a particle were to cease at any instant 
 it would proceed to move with the velocity it then has and in 
 the direction in which it is then moving, i.e. in the tangent 
 to the curve at the point where it was at the instant the force 
 ceased. , If then the particle continues to pursue a curvilinear 
 
 a 
 
 path,' the value of the expression - at any point measures 
 
 the acceleration in direction of the normal, which must operate 
 upon the particle in order tp deflect it from the tangent and 
 
NORMAL ACCELERATION. 249 
 
 retain it in its curvilinear path. This accelerating force in 
 direction of the normal has been frequently called the cenPri- 
 fugal force of the particle, — ^vaguely conveying an impression 
 as it were that the particle of itself resisted curvilinear motion 
 and exerted a force per se to move in a rectilinear path, which 
 innate tendency was only overcome by the action of some 
 external force; whereas the dynamical principles now univer- 
 sally accepted, teach us that a particle of matter exerts no 
 force upon itself, but submits passively to the action of any 
 external force ; retaining whatever motion has been impressed 
 upon it till it is modified by the action of some new force. 
 We would recommend the student to avoid this vague use of 
 the term centrifugal fcyrce, or if he uses it at all, to use it 
 simply as an equivalent for the force in direction of the nor- 
 mal, viz. — or ; according as he is estimating the accele- 
 
 r P 
 
 rating or moving force in that direction (m being the mass of 
 the particle). 
 
 107. A particle moves on a cycloid whose axis is vertical 
 under the action of gravity ; to find the pressure on the curve, 
 (Fig. Art. 100.) 
 
 Let m be the mass of the particle moving on the concave 
 side of the curve, B the reaction or pressure which the 
 material curve exerts on the particle towards the concavity, 
 which is consequently equal .to the pressure which the par- 
 
 7? 
 
 tide exerts on the curve in the opposite direction, then — 
 
 will be the accelerating force of this pressure ; also let ^ be 
 the angle which the normal PQ makes with the vertical j then 
 since gravity acts downwards, g cos ^ will be the resolved 
 part ,of the accelerating force of gravity estimated in direction 
 
250 PRESSURE ON A CURVE. 
 
 QP, and ^ cos ^ will be the whole actual acceleration in 
 
 direction of the normal. 
 
 a 
 
 But — measures the acceleration necessary to make the 
 P 
 particle move on the curve as it actually does (Art. 105). 
 
 These two expressions then must be equivalent, and we 
 
 shall have 
 
 — = a cos d> : 
 
 p m ^ ^' 
 
 .'. Ii = mi — \- ff cos (j) I , 
 
 XfJ J 
 
 the required expression for the pressure. 
 
 Cor. 1. If the particle describe a cycloid by being 
 attached to a string, as in Art. (102, Cor. 3), the tension of the 
 string on the particle must be the same as the pressure of the 
 curve in the previous Article, i.e. tension of the string in any 
 
 position EQP— mi— + gcosj)\. 
 
 Cor. 2. If a particle move on any curve under the action 
 of any force, and 8 be the resolved accelerating force in 
 direction of the normal, estimated positive towards the con- 
 cavity, we should get by the same reasoning as in the 
 
 present Article, B = m{ 8\. 
 
 \p J 
 
 Cor. 3. Since the curve can only exercise a pushing 
 force upon the particle, if the expression for H becomes nega- 
 tive in any case (which indicates that the curve ought to 
 exercise a pulling force) the particle will quit the curve, — and 
 moreover will quit it at the point where ^ is = 0, provided 
 
PKOBLEMS, 251 
 
 that as the particle passes through that point the expression 
 for B changes sign from positive to negative. 
 
 If the particle be moving in a tube of very small bore, 
 instead of on a curve simply, the direction of the pressure 
 which the tube exerts upon the particle will change its 
 direction at such a point as is here contemplated, i.e. if 
 when the particle is on one side of the point the pressure acts 
 towards the concave side of the curve, when the particle is 
 on the other side of the point the pressure will act towards the 
 convex side of the curve, and vice versd. 
 
 108. We will illustrate the principles of this chapter by 
 the following problems. 
 
 Pkob. a particle descends down the arc of a smooth ver- 
 tical circle, starting from rest at the vertex ; to find where ike 
 particle, will quit the circle. 
 
 Let V be the velocity of the particle 
 in any position P in its course down 
 the circle. 
 
 AO the vertical diameter, the 
 centre of the circle whose radius = a. 
 
 PN horizontal, POA = 0, S the 
 pressure of the circle on the particle 
 outwards from 0. 
 
 Then v''==2g, AN since the particle starts from rest at J, 
 
 and since the radius of curvature is the same at every point, 
 
 v" 
 and = a, therefore — measures the acceleration at P in direc- 
 a 
 
 tion PO: but ^ cos 6 is the resolved part of gravity in direction 
 
 jf 
 
 PO, and therefore qcosd is the whole actual acceleration 
 
 on the particle in direction PO ; 
 
252 PROBLEMS. 
 
 ,-. -=q COS0 ; 
 
 a " m 
 
 .-. R = 'm (g cos 9 j , 
 
 aud v" = 2gAN= 2ga (1 - cos 0) ; , 
 
 .-. E = mg{3 cos ^-2). 
 This gives the pressure at any point P, and so long as cos 6 
 is > - , i? is positive and the particle remains in contact with 
 
 o 
 
 2 
 
 the curve; but when 6 becomes so large that cos ^ < o ' t^^n 
 
 B becomes negative, and it would re(Juire the curve to exert 
 
 a pulling force in order to retain the particle in contact with 
 
 2 
 it. Hence at the point where cos ^ = - , B changes sign from 
 
 positive to negative, and the particle quits the curve. 
 
 2 1 
 
 At the point where cos ^ = - , AN= ,-40. 
 
 After quitting the curve the particle proceeds to describe 
 a parabola. 
 
 109. PrOB. a particle is whirled round in a vertical plane, 
 being attached to one end of an inelastic string, the other end of 
 which is fixed, — to find the tension of the string in any position, 
 and the conditions that the particle may describe a complete 
 circle. 
 
 Let (Fig. p. 251) be the fixed end of the string whose 
 
 length is = a, P the position of the particle when the string PO 
 
 makes any angle 6 with the vertical OA, draw PN horizontal, 
 
 then 
 
 AN= a (1 - cos'^). 
 
PEOBLEMS. 253 
 
 Let The the tension of the string when the particle is at P; 
 u, V the velocity when it is at A, P respectively ; then 
 
 v' = m' + 2g . ^iV=M°+ 2ffa (1 - cos 6). 
 
 T 
 
 Also — = accelerating force of the tension of the string in 
 
 direction PO, 
 
 g cos 6 = resolved part of gi*avity in direction PO ; 
 
 T 
 .". — Vq cos d = whole acceleration in PO, 
 
 = ^\ by Art. (107), 
 = — + 2,7 (1 - cos 6) ; 
 
 .-. r= ml- +^(2 -3 cos (9) J. 
 
 This gives the tension of the string in any position. 
 
 T is least when cos 6 = 1; i.e. when 6 = or P is at A, 
 and increases continually as 6 increases, till when 6 = Tr (or 
 the particle is at the lowest point), Tis greatest. 
 
 In order that the particle may describe a complete circle, 
 the tension must never be negative, otherwise the string would , 
 become slack. 
 
 If we make the least value of Tzero, i. e. put t=0 when 
 6 = 0, -we, get 
 
 m" 
 
 — +^ (2 - 3) = 0, or u^ = ga, or m = i^ijga) ; 
 
 Of 
 
 which , expresses the least velocity the particle may have at 
 the highest point in order to describe a complete circle. 
 
254 NEWTON'S MODE OP 
 
 The greatest velocity is at the lowest point, and if 
 
 the greatest Telocity = tjipga). 
 
 The expression for the tension in this case becomes 
 
 T=Zmg{l-CQ&e), 
 
 the maximum value of which is when cos ^ = — 1, or when 
 the particle is at the lowest point ; the tension is then ec[ual 
 to ^mg = 6 . weight of the particle. 
 
 The conditions necessary to be fulfilled in order that a 
 complete circle may be described are 
 
 (i) the velocity at the lowest point must not be < \/{5ga). 
 
 (ii) the string must be capable of sustaining a strain 
 equal to at least six times the weight of the particle. 
 
 110. We will conclude this chapter with a short account 
 of the method employed by Newton to determine the elas- 
 ticity of different substances. 
 
 Let A, B be two balls suspended from fixed points C, D 
 by parallel strings, so that they 
 may be in contact at the extremi- 
 ties of horizontal diameters. If the 
 balls be drawn aside through given v 
 arcs, the velocities with which they "'x^ 
 strike each other can be found \a1I\1b\ 
 
 (Art. 97, Cor. 3), and by a proper 
 arrangement of these arcs they can be made to impinge upon 
 each other when they are in their lowest position. By ob- 
 serving the arcs through which they rebound, the velocities 
 with which they separate after the impact can be obtained, 
 and thence the coefficient of elasticity. 
 
 
MEASUEING ELASTICITY. 255 
 
 By experiments of this kind Newton determined the co- 
 efficient (e) of elasticity of balls of worsted to be about - , — of 
 
 balls of steel it was nearly the same, — of cork a little less, — of 
 
 8 15 ' 
 
 ivory e = - , — and of glass e = ~. See Principia, Bk. i. Scho- 
 lium to the Laws of Motion ; where Newton further shews 
 how allowance may be made for errors arising from the resist- 
 ance of the air. 
 
 Again, if B be drawn aside and allowed to impinge upon 
 A at rest, the velocities of each after impact will be found to 
 be the same as result from the principles assumed in the 
 chapter on collision. 
 
 Or again, suppose the balls to be of wood, and let one of 
 them B have a small steel point projecting from it which 
 would cause it to stick to A after the impact, — by properly 
 adjusting the arcs through which the balls are displaced their 
 velocities at impact can be made to be inversely proportional 
 to their masses, and by loading one of them with lead their 
 masses can be made to bear any proportion, — it will be found 
 that they remain at rest after the impact, shewing that equal 
 momenta in opposite directions destroy each other. 
 
( 256 ) 
 PROBLEMS AND EXAMPLES. 
 
 EXAMPLES NOT INVOLVING PEICTION. CHAPTERS I. II. 
 
 1. Two given forces act at a point ; if the angle between 
 their directions be increased, the magnitude of their resultant 
 will be diminished, and. vice versd. 
 
 2. Three given forces cannot be made to balance each 
 other by any arrangement of their directions, if the sum of 
 any two be less than the third. 
 
 3. Two equal forces applied at a given point have a. 
 resultant given in magnitude and direction,^find the locus 
 of the extremity of the straight line which represents either 
 force. 
 
 4. If be a point within a triangle ABC, and D, E, F 
 the middle points of the sides, — the system of forces repre- 
 sented by 0^, OB, OG will be equivalent to those repre- 
 sented by OD, OE, OF. 
 
 5. A circular hoop is supported in a horizontal position, 
 and three weights P, Q, R are suspended over its circumfer- 
 ence by three strings meeting in the centre; what must be 
 their positions so that they may balance each other ? 
 
 The angle between the directions of any two strings will be given by the 
 fonnulse of Art. 23. 
 
 6. The angles A, B, oi a, triangle are 30°, 60°, 90" 
 respectively. • The point C is acted on by forces in directions 
 
FORCES IN ONE PLANE. CHAPS. I. 11, 257 
 
 CA, CB inversely proportional to CA, CB. Find the magni- 
 tude and direction of their resultalit. 
 
 Result. The resultant makes an angle 60° with CA and its magnitude 
 : force in C^ :: AB: CB. 
 
 7. If a point be acted on hj three forces parallel and 
 proportional to the three sides AB, BC^DG oi a quadrilateral, 
 shew that the resultant of the forces is represented in magni- 
 tude and direction by EGE', E being the middle point of 
 AD, and CE' being eqUal to EG. 
 
 8. If two forces P and Q act at such an angle that 
 B= P, shew that if P be doubled, the new resultant will be 
 at right angles to Q. 
 
 9. The resultant of two forces P and Q acting on a 
 particle is the same when their directions are inclined at an 
 
 Z.0 as when they are inclined at an z - — 5 to each other : — 
 
 shew that tan 6 = J2 — 1. 
 
 10. A uniform sphere moveable about a fixed point in 
 
 its surface, rests against an inclined plane ; find the pressure 
 
 on the fixed point. 
 
 Remit. If a be the inclination of the plane and ^ the angle which the 
 radius to the fixed point makes with the vertical, 
 
 sino • i^ ^ T 
 
 pressure = -. — ; -pr. . wennht of sphere. 
 
 ^ 3m(a+/3) ^ ^ ^ 
 
 11. Two equal weights P, Q are connected by a string 
 which passes over two smooth pegs A, B situated in a hori- 
 zontal line, and supports a weight W which hangs from 
 a smooth ring, through which the string passes. Find the 
 position of equilibrium. 
 
 Remit. The depth of the ring below the line AB 
 
 W 
 t: ===^= . length AB. 
 
 P. M. 17 
 
258 .. PEOBLEMS. 
 
 12. The resultant of two forces P, Q acting at an angle' 
 6 is equal to {2m, + 1) V (-P" + Q") ; when they act at an angle 
 
 ^ - e, it is equal to (2to- 1) V(P' + Q') ; shew that 
 tan 6 = 
 
 m + 1 
 
 13. Six forces in one plane represented in magnitude 
 and direction by the lines OA, OB, 00, O'A, O'JB, O'O, 
 when applied at a point, balance each other. Prove that the 
 algebraical sum of the triangles OBO, O'BO (considered of 
 different signs when 0, 0' are on opposite sides of BO) is 
 equal to two-thirds of the triangle ABO. 
 
 a * « * * * * 
 
 14. A rod 5 feet long has a string 7 feet long attached 
 to its ends, and by this it is hung over a small smooth 
 fixed peg, so that the parts of the string are as 4 : 3. Find 
 the position of the centre of gravity of the rod and the 
 pressure on the peg. 
 
 12 [^ 
 Sesult. Depth of centre of gravity of tlie rod below the peg = — ^^ , in- 
 
 • 7 
 
 clination of the rod to the vertical = sin""' — =, pressure on peg = weight of 
 
 rod. 
 
 15. A smooth circular ring is fixed in a horizontal posi- 
 tion, and a small ring sliding upon it is in equilibrium when 
 acted on by two strings in the direction of the chords PA, 
 PB; shew that if PC be a diameter of the circle the tensions 
 of the strings are in the ratio of 5(7 to AO. 
 
 16. Three forces P, Q, R acting upon a point and keep- 
 ing it at rest, are represented by lines drawn from that point. 
 If P be given in magnitude and directioii, and Q in magni- 
 
FORCES IN ONE PLANE. CHAPS. I. II. 259 
 
 tude only, find the locus of the extremity of the line which 
 represents the third force B. 
 
 17. At any number of points of a parabola forces are 
 applied, represented by the tangents and normals at those 
 points, — shew that the parabola will remain at rest if the 
 focus is fixed^ 
 
 18. A circular disc is kept at rest "by three forces acting 
 perpendicularly to the circumference at three given points 
 therein ; shew that the forces are as the sides of the circum- 
 scribing triangle that pass through those points. 
 
 19. R being the resultant of P and Q, let B be equal to 
 VS . Q, and make an angle of 30° with P; — find P in terms 
 of. Q.. 
 
 Remit. P=Q or P=2Q. 
 
 20. AB is a uniform rod, — of weight W, — moveable in 
 a vertical plane about a hinge A ; a given weight P sustains 
 the rod by means of a string BOP passing over a smooth 
 pin C, situated in the vertical through A and at a distance 
 AC=AB. In the oblique position of equilibrium of the rod, 
 
 p 
 
 ^ ACB = cos~^-jrr- 
 W 
 
 21. Two rods similar in every respect — (the weight of 
 each being W) — are capable of motion in a vertical plane 
 round a common fixed pivot at one extremity of each, and 
 they are kept in equilibrium in a position inclined at ^ ^ to 
 the horizon by a string placed over the other ends and kept 
 stretched by two equal weights (P, P) at its extremities. 
 
 Shew that 
 
 2P+W 
 
 tan^, = : 2^ 
 
 17—2 
 
260 . PEOBLEMS. 
 
 22. ABDG is a quadrilateral, and is acted on by forces 
 whicli act In the direction of, and are proportional to, AB, A C, 
 DB, DC respectively; shew that their resultant is parallel 
 and proportional to the line joining the middle points of the 
 diagonals, 
 
 23. A lever without weight in the form of the arc 2a of 
 a circle subtending an ^ 2a at its centre, having two weights 
 P and Q suspended from its extremities, rests with its con- 
 vexity downwards upon a horizontal plane; determine the 
 position of equilibrium. 
 
 Bemlt, The chord PQ, is inclined to the horizon at an angle 
 
 tan~i 
 
 (J^tana). 
 
 24. The ends of a uniform heavy rod are connected, by in- 
 extensible strings without weight, with the ends of another 
 uniform rod which is moveable about Its middle point. Prove 
 that,. when the system Is in equilibrium, either the rods or 
 the strings are parallel. 
 
 25. If a uniform heavy rod be supported by a string 
 fastened at its ends, and passing over a smooth peg ; prove 
 that it can only rest in a horizontal or vertical position. 
 
 26. Two equal circles intersect in A and B'. any line 
 PQN perpendicular to AB meets the circles in P and Q and 
 AB in N. Prove that the resultant of four forces repre- 
 sented by PA, PB, QA, QB is of constant magnitude. 
 
 ******* 
 
 27. Explain how a vessel Is enabled to sail in a direction 
 nearly opposite to that of the wind. 
 
 28. Explain how the force of the current may be taken 
 advantage of to urge a ferry-boat across a river ; the centre of 
 
FORCES IN ONE PLANE. CHAPS. I. II. 261 
 
 the boat being attached, by means of a long rope, to a moor- 
 ing in the middle of the stream. 
 
 29. The whole length of each oar of a boat is 10 feet, 
 and from the hand to the rowlock the distance is 2 ft. 6 in. ; 
 each of eight men sitting in the boat pulls his oar with a force 
 of 54 lbs. Supposing the blades of the oars not to move 
 through the water, find the resultant force propelling the boat. 
 
 Semlt. Propelling force=1441ba. 
 
 30. At what height from the base of a pillar must the 
 end of a rope of given length be fixed, so that a given power 
 acting at the other end may be most efiectually exerted to 
 overturn the pillar? 
 
 Ilesv.lt, —— . length of rope. 
 
 31. A uniform beam of length 2a rests against a vertical 
 plane and over a peg at a distance h from the plane ; shew 
 
 that the inclination of the beam to the vertical is 
 
 =-7©' 
 
 32. A uniform rod whose weight is W is supported by 
 two fine strings (one attached at either end), which passing- 
 over small fixed smooth puUies carry weights w^ , w^ respec- 
 tively. Shew that the inclination of the rod to the horizon 
 is 
 
 w^~w^ 
 
 33. Two equal uniform heavy straight rods are con- 
 nected at one extremity by a string, and rest upon two 
 smooth pegs in the same horizontal line, one rod upon one 
 peg, and the other upon the other: — the distance between 
 the pegs being equal to the length of each rod, and the 
 
262 PEOBLEMS. 
 
 length of the string heing half the same : shew that the 
 rods rest at an angle 9 to the horizon, such that 
 2cos'^=l. 
 
 34. A string is knotted so as to form an equilateral 
 triangle, and is placed symmetrically within another equi- 
 lateral triangle nine times as great, each knot being con- 
 nected with the two nearest angles of this triangle by strings 
 solicited with a tension P. If T be the tension of the tri- 
 angular string, then will P= Tn^X. 
 
 35. Two straight lines AB, A C make an ^ 2a with 
 each other: -When a certain force M is resolved into two 
 forces parallel and perpendicular to AB, P is the component 
 parallel to AB; similarly, when B is resolved into two 
 forces parallel and perpendicular to AG, Q is the compo- 
 nent parallel to A 0, — shew that 
 
 i2 = 4 {(P+ e)=sec^a+ (P- (3)'cosec''a)*, 
 
 and that the direction of B makes an angle 
 
 jP-Q \ 
 tan ( T> , f) cot a I 
 
 with the straight line bisecting the ^ BA G. 
 
 36. Two equal weights (P, P) are attached at the ex- 
 tremities of a string which passes over three small pullies 
 forming an equilateral triangle; a small heavy ring [W) is 
 slipped over the uppermost pully and descends by its own 
 weight ; find the position of equilibrium. 
 
 jResult, The portions of the string which are not vertical are incline^ to 
 the vertical at an angle 2 siu-^ ( - a/ — J . 
 
 37. A uniform heavy rod of given length is to be sup- 
 ported in a given position, with its upper end resting at a 
 
FOECES IN ONE PLANE. CHAPS. I. II. 263 
 
 given point against a smooth vertical wall, by means of a fine 
 thread attached to the lower end of the rod and to a point in 
 the wall. Find hy a geometrical construction the point in the 
 wall to which the string must be attached. 
 
 38. A flat semicircular board with its plane vertical, and 
 curved edge upwards, rests on a smooth horizontal plane, 
 and is pressed at two given points (P, Q) of its circumference 
 by two beams which slide in smooth vertical tubes; find the 
 ratio of the weights of the beams that the board may be in 
 equilibrium. 
 
 Eesult. If vi, |8 be the angles which the radii at P, Q make with the 
 horizon — then the weight of the beam at P : that at Q=tana : tan/3. 
 
 39. Three equal heavy cylinders, — (weight of each = Tr), — 
 each of which touches the other two, are bound together by 
 a string and laid upon a horizontal plane; the tension (T) 
 of the string being given, find the pressures between the 
 cylinders. 
 
 Result. Pressure between the upper and either of the lower cylinders 
 
 W W 
 = T + — = — ^between two lower cylinders = T = . 
 
 40. Three straight tobacco-pipes rest upon a table, with 
 their bowls mouth-downwards in the angles of an equilateral 
 triangle, the tubes being supported in the air by crossing 
 symmetrically, each under one and over the other, so as to 
 form another equilateral triangle ; shew that the mutual pres- 
 sure of the tubes varies inversely as the side of the latter 
 triangle. 
 
 41. Ji ABO be a right-angled triangle, and ABDE, 
 A CFG be the squares on the sides constructed as in Euclid 
 V. 47, prove that the resultant of forces represented by CD, 
 BF is parallel- to a diagonal of the rectangle whose sides are 
 AE,AG. • ■ 
 
264 PROBLEMS. 
 
 42. An elliptic lamina is acted on at the extremities of 
 pairs of conjugate diameters by forces in its own plane 
 tending outwards and normal to its edge : there will be equi^ 
 librium if the force at the end of every diameter be propor- 
 tional to the conjugate. 
 
 43. Three equal rods are jointed by smooth compass-joints 
 at the extremities so as to form an equilateral triangle. Find 
 the direction of the pressures on the lower joints when the 
 triangle is suspended from one angle. 
 
 Js 
 
 ResuU. They are inclined to the horizon at an angle tan~i -— . 
 
 44. One comer of a square lamina is fixed, and equal 
 forces (P, P) act in order of direction along the two sides 
 which do not pass through that corner. If a single force 
 applied at the centre of the lamina keeps it at rest, deter- 
 mine this force, and the pressure on the fixed point. 
 
 Result. A single force R=2iJiP acting perpendicular to the diagonal 
 passing through the fixed point : and pressure on fixed point =iJzP acting 
 parallel and opposite to R. 
 
 45. A cylindrical shell, without a bottom, stands on a 
 horizontal plane, and two smooth spheres are placed within it, 
 whose diameters are each less whilst their sum is greater than 
 that of the interior surface of the shell ; shew that the cylinder 
 will not upset if the ratio of its weight to the weight of the 
 upper sphere be greater than 2c—a — b: c, — where a, h, c are 
 the radii of the spheres and cylinder. 
 
 46. Two forces in the ratio 1 + n: 1 where n is small, 
 act upon a point in directions inclined at an angle a ; shew 
 that the sine of the angle which the direction of the resultant 
 
 makes with that of the larger force = f 1 — - j sin - nearly. 
 
FORCES IN ONE PLANE. CHAPS. I, II, 565 
 
 47. An endless string supports a system of equal heavy 
 puUies, the highest one of which is fixed, the string passing 
 round every puUy and crossing itself between each. If 
 a, /3, 7, &c. be the inclinations to the vertical of the successive 
 rectilinear portions of string, prove that cos a, cosyS, cos 7, 
 &c. are in arithmetic progression. 
 
 48. A heavy rod — (weight W, length 2a) — can turn freely 
 about a hinge at one extremity A; and it carries a heavy ring 
 (P) which is attached to a fixed point C in the same hori- 
 zontal plane with the hinge, by means of a string of length 
 (c) equal to the distance between the point and the hinge. 
 The z which the rod makes with the horizon in the position 
 of equilibrium is defined by the equation 
 
 Wa 
 cos 2d + —p- cos 6 = 0, 
 
 49. A sphere of weight W is moveable about a point in 
 Its circumference, at which a string is attached which passes 
 over the sphere and supports a weight P; shew that the 
 diameter of the sphere which passes through the point of 
 suspension is inclined to the vertical at an angle 
 
 50. In a triangular lamina ABC, AD, BE, CF are the 
 perpendiculars on the sides, and forces represented by the 
 lines BD, CD, CE, AE, AF, BF, are applied to the lamina ; 
 prove that their resultant will pass through the centre of the 
 circle described about the triangle. 
 
 51. Two uniform rods AB, BG of similar material are 
 connected by a smooth hinge at B, and have smooth rings at 
 their other extremities, which slide upon a fixed horizontal 
 
266. , PPOBLEMS. 
 
 wire : shew that the only positions of equilibrium are, those 
 in which the lesser rod is vertical. 
 
 52. Two small rings without weight slide on the arc of 
 a smooth vertical circle, — a string passes through both rings, 
 and has three equal weights attached to it, one at each end 
 and one between the rings ^ — ^in the position of equilibrium 
 the distance between the rings is equal to the radius of the 
 circle. ' 
 
 53. A ring whose weight is P, is moveable along a 
 smooth rod inclined to the horizon at an angle a, another 
 ring of weight F' is moveable along another rod in the same 
 vertical plane as the former, and inclined at an angle a' to the 
 horizon ; a string which connects these rings passes through 
 a third ring of weight 2 W which hangs freely ; shew that the 
 system cannot be in .equilibrium unless 
 
 P tan a - P' tan a' +. W (tan a - tan a') .= 0., 
 
 54. A square rests with its plane perpendicular to a 
 smooth wall, one corner being attached to a point in the 
 wall by a string whose length is equal to a side of the square ; 
 shew that the distances of three of its angular points from 
 the wall are as 1 : 3 : 4. 
 
 55. A uniform square board is capable of motion in a 
 vertical plane about a hinge A at one of its angular points ; a 
 string attached to G one of the nearest angulai- points and 
 passing over a pully vertically above the hinge, at a distance 
 from it equal to a side of the square, supports a weight whose 
 ratio to the weight of the board is 1 : V(2). Find the posi- 
 tions of equilibrium. 
 
 ResuU. A C makes with the vertical an 1 30° or / 90". 
 
 56. One end of a beam whose weight is W, is placed on? 
 a smooth horizontal plane ; the other end, to which a, string ia 
 
FOKCES IN ONE PLANE. CHAPS, I. II. 267 
 
 fastened, Tests against another smooth plane inclined at an 
 angle a to the horizon ; the string passing over a puUy at the, 
 top of the inclined plane hangs vertically, supporting a weight 
 P. Shew that the beam will rest in all positions if 
 
 2P= IF sin a. 
 
 57. Two equal circular discs — of radius r — with smooth 
 edges are placed on their flat sides in the corner between two 
 smooth vertical planes inclined at ':2a, and touch each other in 
 the line bisecting the angle ; the radius of the least disc which 
 may be pressed between them without causing them to sepa,rate 
 
 1 — cos a. 
 
 = r , 
 
 cos a 
 
 58. One end of a string is fixed to the extremity of a 
 smooth uniform rod, and the other to a ring without weight 
 which passes over the rod, and the string is hung over a 
 smooth peg. Determine the least length of the string for 
 which equilibrium is possible, and shew that the inclination 
 of the rod to the vertical cannot be less than 45°. 
 
 59. Two similar and equal smooth rods AB, BO, have 
 a compass-joint at P; a ring without weight slides on BG, 
 being attached to ^ by a string, so that the rods can rest with 
 their ends on a smooth horizontal plane. Shew that the 
 mutual pressure at B is perpendicular to BG. 
 
 60. Shew that the moment of a force represented by AB 
 about any line passing 'through a point P will be represented 
 by double the projection of the triangle PAB on a plane per- 
 pendicular to the line. 
 
 Prove by this method of projection — or otherwise — that 
 the sum of the moments of two forces (whose lines of action 
 
268 PEOBLEMS. 
 
 intersect) about any line is equal to the moment of their re- 
 sultant about the same line. 
 
 61. The sides of a rhombus ABOD are hinged together 
 at the angles ; sX A, C are two pulling forces (P, P) acting 
 in the diagonal A G ; and at B, D there are two other pulling 
 forces ( Q, Q) acting in BD ; shew that 
 
 .DAB = cos-f^). 
 
 62. If the parallelogram of forces be true for any two 
 forces making a given angle with each other, prove that it 
 will be also true for any two forces making any other angle 
 with each other. 
 
 63. A particle Pis placed in a smooth horizontal tube AB, 
 and is acted on by two forces tending to two fixed points C, D, 
 and proportional to the distances CP, PD; find the force 
 necessary to keep P at rest in a given position. 
 
 64. Two equal heavy beams AB, OB are connected 
 diagonally by similar and equal elastic strings AB, BG, — 
 determine the position of equilibrium when AB is held hori- 
 zontal : and shew that if the natural length of each string 
 equals AB, and the elasticity be such that the weight of AB 
 would stretch the string to 3 times its natural length, then 
 
 AB~ BG^ AG' 
 
 65. A small smooth ring is capable of sliding on a fine 
 elliptic wire, whose transverse axis is vertical; two strings 
 attached to the ring pass through small smooth rings at the 
 foci and sustain given weights : shew that if the ring be in 
 
FORCES IN ONE PLANE. CHAPS. I. II. 269 
 
 equilibrium at any point, besides the highest and lowest 
 points of the wire, it will be in equilibrium in every position. 
 
 66. Two equal rods AB, A C without weight are con- 
 nected by a hinge at A and are placed in a vertical plane 
 resting dn a smooth sphere so that the point A is vertically 
 over the centre 0. A heavy ring slides on a string attached 
 to the two ends B and C, the length of the string being twice 
 that of either rod. If BB be the perpendicular drawn from 
 B on AO produced, prove that in the position of equilibrium 
 A . AB = 2 BB^ : — supposing the sphere to be so small that 
 the string is clear of it. 
 
 67. A small ring (weight W^ is moveable on a rod 
 whose inclination to the horizon is a^ ; another ring (weight 
 PFj) is moveable on another rod in the same vertical plane, 
 whose inclination is a, ; a slender thread connecting the rings 
 carries a ring (weight W). Shew that 
 
 tan g, _ W+ 2 W^ 
 tana," W-ViW,' 
 
 68. Forces are applied at the middle points of the sides 
 of a rigid plane polygon, perpendicular to the sides, and pro- 
 portional to them in magnitude, all the forces tending in- 
 wards <yr all outwards ; shew that the system of forces is in 
 equilibrium, 
 
 69. Two rods of equal uniform thickness have their ends 
 joined by a compass-joint and rest in a circle whose plane is 
 vertical ; prove that if the rods are at right angles they are 
 equally inclined to the horizon. 
 
 70. A smooth heavy rod AB — weight TF— moveable in 
 a vertical plane about a hinge at A, leans against a heavy 
 
270 PROBLEMS. 
 
 prop CD — weiglit P — also moveable in the same plane about 
 a hinge at G. Find the position of equilibrium. 
 
 Result If AB=2a, CD='2]), CA = c, lBAO=B, LT)OA = <p, we shall have 
 c sin e= 25 sin (9 + ^) and alf sin 2^008 (ff + 0) + P6 sin 20=0. 
 
 71. A cylinder — (length=A, rad. base=a, ■weight = TF) — 
 rests with its base on a smooth inclined plane — (inclination 
 = a) J — a string attached to its highest point, passing over a 
 pully at the top of the inclined plane, hangs vertically and 
 supports a weight P ; the portion of the string between the 
 cylinder and pully is horizontal : determine completely the 
 conditions of equilibrium. 
 
 Result. We. must have P=W tsi,na, with the condition that tana is 
 
 2o 
 not > -T- • 
 
 72. A cylinder with its base resting against a smooth 
 vertical plane is held up by a string fastened to it at a point 
 of its curved surface whose distance from the vertical plane 
 is h. Shew that h must be > J — 2a tan and < b ; where 
 2b is the altitude of the cylinder, a the radius of the base, 
 and 9 the angle which the string makes with the vertical.^ 
 
 73. Four rods jointed at their extremities form a quadri- 
 lateral which may be inscribed in a circle ; if they be kept in 
 equilibrium by two strings joining the opposite angular points, 
 shew that the tension of each string is inversely proportional 
 to its length. 
 
 74. A regular hexagon composed of six equal heavy, 
 rods, moveable about their angular points, is suspended from 
 one angle, which is connected by threads with each of the 
 opposite angles. The tensions of the threads are as t/3 : 2, 
 
 75. A string 9 feet long has one end attached to the 
 extremity of a smooth uniform heavy rod two feet in length; 
 
FOECES IN ONE PLANE. CHAPS. I. II. 271 
 
 and at the other end carries a ring which slides upon the rod. 
 The rod is suspended by means of the string from a smooth 
 peg ; prove that if 6 be the angle which the rod makes with 
 the horizon, then tan 6 = 3~^ — 3"^. 
 
 76. If two forces acting along chords of a circle are 
 inversely proportional to the lengths of the chords, their 
 resultant will pass through one or other of the points of 
 intersection of lines drawn through the extremities of the 
 chords. 
 
 77. A thread passing over a vertical hoop is held to the 
 hoop by two equal rings Pj, P^, and a third equal ring 
 Pj hangs on the thread between the two ; prove that if Q be 
 the point in which a tangent to the hoop parallel to P^ P^ 
 meets the vertical through P^, thenP^ is situated at the centre 
 of gravity of the triangle Pj QP,^. 
 
 78. If lines be drawn from any point whatever to four 
 fixed points in the same plane with it, and these lines repre- 
 sent forces all acting from or all towards the point ; shew 
 that their resultant will pass through a certain fixed point 
 and will be proportional to the distance of the first fixed 
 point from it, 
 
 * 79. Three uniform beams AB, BG, CD, of the same 
 
 thickness and of lengths I, 21, I respectively, are connected by 
 
 hinges at B and G, and rest on a perfectly smooth sphere, the 
 
 radius of which = 21, so that the middle point of BO and the 
 
 extremities of A, D are in contact with the sphere; shew that 
 
 91 
 the pressure at the middle point of 5(7= r^ of the weight of 
 
 the beams. ; 
 
272 PEOBLEMS. 
 
 80. Three forces act in equilibrium at the angles of a tri- 
 angle, one bisecting the angle at which it acts, and the other 
 two making equal angles with the side opposite to that angle; 
 shew that the forces are as the sides opposite to their points of 
 application, and that they will balance if turned through any 
 equal angles in the same direction. 
 
 81. A quadrilateral is formed by four rigid rods jointed 
 at the ends ; shew that two of its sides must be parallel, in 
 order that it may presei-ve its form when the middle points of 
 either pair of opposite sides are joined together by a string in 
 a state of tension. 
 
 82. A cube whose weight is W rests upon a horizontal 
 table, and is cut by three planes passing through a diagonal 
 of the upper face and the several corners of the lower face. 
 If the parts cut off be placed together again, their faces being 
 supposed perfectly smooth, and be kept in equilibrium by a 
 horizontal string tied round the cube, — ^prove that the tension 
 of the string is ^ W. 
 
 83. A uniform beam PQ of given weight ( W) and length 
 rests in contact with a fixed vertical circle, whose vertical 
 diameter is AB, in such a manner that strings AP, BQ at- 
 tached to the rod and circle are tangents to the circle at the 
 points A and B. Find the tensions of the striflgs, and shew 
 that the conditions of the problem require that the inclination 
 of the beam to the vertical must be 
 
 < sm ' 
 
 Result. If a = inclination of beam to the vertical, the tensions of the strings 
 are ^W{cot a±sec o). 
 
 84. A particle is placed on a smooth square table at 
 distances c^, c^, c^, c^ from the comers, and to it are attached. 
 
F0ECE8 IN ONE PLANE. CHAPS. I. II. 273 
 
 strings passing over smooth pullies at the comers, and sup- 
 porting weights Pj, Pj, P3, Pii shew that if there is equi- 
 librium, 
 
 a being a side of the table. 
 
 85. A cone of given weight W is placed with its base 
 on a smooth inclined plane (a), and supported by a weight 
 
 W which hangs by a string fastened to the vertex of the 
 cone, and passing over a puUy in the inclined plane at the 
 same height as the vertex. Find the angle (2/3) of the cone 
 when the ratio of the weights is such that a small increase of 
 
 W would cause the cone to turn about the highest point of 
 
 the base, as well as slide. 
 
 3 
 Eesult. tan /3 == ^ sin 2a. 
 o 
 
 86. A cone, the vertical angle of which is 2 tan"'|, 
 rests with its vertex against a smooth vertical wall, a point 
 in its base being attached to a point in the wall by a string 
 to which the axis of the cone is parallel when it is in equi- 
 librium ; shew that the tension of the string is W'^2, and 
 that the distance of thevertex of the cone from the fixed 
 
 point in the wall is — - — : — where W is the weight of the 
 
 cone, and h the length of its axis. 
 
 87. A bowl is formed from a hollow sphere of radius 
 a: it is so placed that the radius of the sphere drawn to 
 each point in the rim makes an ^a with the vertical, and 
 the radius drawn to a point A of the bowl makes an ^y8 
 with the vertical; — ^if a smooth uniform rod remains at rest 
 
 p. M. 18 
 
274 PKOBLEMS. 
 
 when placed with one extremity at A, and with a point in 
 its length on the rim of the bowl, shew that the length of 
 
 the rod is = ia sin yS sec — - — . 
 
 FKICTION. CHAP. III. 
 
 1. Two parallel vertical walls — at a distance c — are one 
 smooth and the other rough, and between them is supported 
 a hemisphere — radius a and weight W- — with its curved 
 surface in contact with the smooth wall, and a point in its rim 
 in contact with the rough wall; the pressure on each wall 
 
 = -W, and the least coefficient of friction consistent with 
 
 .,., . V2ac - c" 
 
 equihbrium = . . 
 
 c-a + |V 2ac - c" 
 
 2. Two rough bodies rest on an inclined plane in a, prin- 
 cipal plane, and are connected by a string which is parallel to 
 the plane ; if the coefficient of friction be not the same for 
 both, find the greatest inclination (a) of the plane which is 
 consistent with equilibrium. 
 
 Jtesult. If W, W be the weights, /i, // their coefficients of friction, the 
 
 value of o=tau ^r+W' " 
 
 3. A uniform ladder 10 ft. long rests with one end against 
 a smooth vertical wall and the other on the ground, the 
 coefficient of friction being = ^; find how high a man (whose 
 weight is 4 times that of the ladder) may rise before it begins 
 to slip, the foot of the ladder being 6 feet from the wall. 
 
 Mesult. ir^ feet, along the ladder. 
 
FRICTION. CHAP. III. 275 
 
 4. A uniform and straight plank — length 2a — ^i-ests witli 
 its middle point upon a rough horizontal cylinder — ^radius c 
 — which is fixed, their directions being perpendicular to each 
 other. Find the greatest weight that can be put upon one 
 end of the plank without its sliding off the cylinder. 
 
 JResuM. P ^f ^, . weight of plank. 
 
 a-e tan" V ° ^ 
 
 5. At what angle of inclination should the traces be 
 attached to a sledge, that it may be drawn up a given hill 
 with the least exertion? 
 
 BesuU. The inclination of the traces to the hill=tan~V- 
 
 6. A string fixed to a point in a rough vertical wall is 
 wrapped round a ball, which is then allowed to hang in con- 
 tact with the wall ; determine the limiting positions of equi- 
 librium. Find the coefficient of friction so that it may be 
 possible for the position of the string not in contact with the 
 ball to be horizontal. 
 
 Besult. The angle the string makes with the wall=6in~^ — . For the latter 
 
 /I 
 
 part of the question /*=!. 
 
 7. Two uniform rods of equal weight AB, BO are in 
 a vertical plane and connected by a firee joint at B ; the point 
 A is fixed and C can move on a rough horizontal plane pass- 
 ing through A; if \ be the / of friction and 6, ^ the angles 
 which the rods make with the vertical when on the point of 
 sliding, 
 
 cot ^ — 3 cot ^ = 2 cot X. 
 
 8. P, Q are two pegs, G the centre of gravity of a slender 
 rod passing over the nearer peg P and under the farther peg Q, 
 ^nd just kept from sliding in direction of its length by the 
 friction between it and the pegs. Find the ratio of PO to PQ 
 
 18—2 
 
276 PROBLEMS. 
 
 in terms of fi, the coefficient of friction between the rod and 
 pegs, and of a the inclination of the rod to the vertical, 
 Ee»wU. PG: PQ=coia-n: 2/t. 
 
 9. A ladder rests against a vertical wall, and is prevented 
 from sliding by the friction of the ground and wall ; shew that 
 when the ladder is on the point of sliding down in a vertical 
 plane, 
 
 tangent of inclination to horizon = — -. — ^-77- , 
 ° fj,{a + b)' 
 
 where fi, jji are the coefficients of friction of the ground and 
 wall, and a, b the distances of the foot and top of the ladder 
 from its centre of gravity. 
 
 Will the extreme angle of inclination be increased or 
 diminished if a man stand on the ladder ? 
 
 10. A heavy hoop — weight W — which has a string coiled 
 round its circumference and a weight P attached to the free 
 extremity of the string, is hung on a rough horizontal peg ; 
 determine the positions in which it will rest. 
 
 Sesvlt. li $ he the angle which the radius drawn through the peg makes 
 
 with the vertical, then :; ; — 7, — ^Sr ^^^ tan fl=u= coefficient of friction 
 
 actually in operation. 
 
 11. The axis of a rough parabola is vertical, shew that 
 the distance of the extreme points at which a particle will rest 
 under the action of gravity is = /* . latus rectum. 
 
 12. A heavy body is kept at rest on a given inclined 
 plane by a force making a given angle with the plane ; shew 
 that the reaction of the plane, when it is smooth, is a har- 
 monic mean between the normal components of the greatest 
 and least reaction, when it is rough. 
 
FKICTION. CHAP. III. 277 
 
 13. A right cone, the height of which is h, rests with its 
 base on an inclined plane ; when the cone is on the point of 
 sliding the coefficient of friction between the plane and the 
 base of the cone is /*. Shew that the resultant action of the 
 
 plane on the cone then acts at a point distant ^ from the 
 centre of its base. 
 
 14. A square lamina has a string, of length equal to a 
 side, attached to one of the angular points ; the string is also 
 attached to a point in a rough vertical wall, against which the 
 lamina rests ; shew that, the coefficient of friction being unity, 
 the angle which the string makes with the wall lies between 
 
 J and ^ tan"' ^. 
 
 15. One end A. of a heavy rod ABC rests against a rough 
 vertical plane, and a point B of the rod is connected with a 
 point in the plane by a string, the length of which is equal to 
 AB=c; determine the position of equilibrium of the rod, and 
 shew how the direction in which the friction acts depends 
 upon the position of B, 
 
 Sesttlt. The angle 9 which the rod makes with the vertical upwards 
 
 . 2c - a 
 
 = tan ' , 
 
 /m 
 
 and the friction acts up or down according as c> or < - . 
 
 16. A cylinder, with its axis horizontal, is held at rest 
 on an inclined plane (a) by a string coiled round its middle, 
 and then fastened on the plane ; find the conditions of equi- 
 librium, friction being considered. 
 
 If 6 be the angle the string makes with the plane obtain the equation 
 
 ,. , sin o 
 
 cos (e.~a) = cos a, 
 
 /* 
 and discuss it. 
 
278 PEOBLEMS. 
 
 17. A cylinder with its axis horizontal, is supported on 
 a rough inclined plane, by a string coiled round it, which 
 after passing over a smooth fixed puUy supports a weight. m 
 times the weight of the cylinder. Proye. that sin a is.<2m, 
 and that /t {cos a + V2re sin a — sin^a} is > n, where a is the 
 inclination of the plane, and /j, the coefficient of friction. 
 Determine the sign of the radical. 
 
 18. A sphere of radius a is supported on a rough in- 
 clined plane — (friction = fi.) — by a string of length - . at- 
 
 tached to it and to a point in the plane. Prove that the greatest 
 possible elevation of the plane in order that the sphere may 
 rest when the string is a tangent is 2 tan"'/*, and find the 
 tension of the string and the pressure on the. plane in this 
 case. 
 
 19. An elastic string has its ends attached to two points 
 
 on the circumference of a vertical circular wire, the line 
 
 joining them being horizontal and equal to the string's 
 
 natural length and their distance 120°. The string passes 
 
 through a small ring which slides on the wire. Find the 
 
 oblique positions of equilibrium, and shew that there are none 
 
 2 
 if the coefficient of elasticity be not > -.- of the ring's 
 
 weight. 
 
 20. OA, OB are radii of a circular arc AB, the former 
 horizontal and the latter inclined at 60° to OA ; find the co- 
 efficient of friction according as a weight Q at jB is on the point 
 of moving up or down the arc, a weight P being attached to 
 ^ by a string FA Q and hanging freely. 
 
 Remit. In the latter case n=~_ji — in the former u=^ . 
 
feictio:n. chap. hi. 279 
 
 21. A heavy hemisphere rests with its convex surface on 
 a rough inclined plane. Find the greatest possible inclination 
 (a) of the plane. 
 
 g 
 
 Result, a = tan"' - . 
 o 
 
 22. A hoard moveahle ahout a horizontal line in its own 
 plane is supported by resting on a rough sphere which lies on 
 a horizontal table; find the greatest inclination at which 
 the board can rest. 
 
 Result, li 11= coefficient of friction between the board and sphere, 
 
 e 
 
 tan-=;«. 
 
 23. A cylinder is supported on a rough inclined plane, 
 with its axis horizontal, by means of a string which is coiled 
 round it, and is attached to a point in the plane, so that the 
 part uncoiled is horizontal. If a be the angle of the plane 
 and the cylinder be only just supported — shew that the coeffi- 
 cient of friction =tan-, and the resistance of the plane 
 = weight of the cylinder. 
 
 24. A heavy circular tube hangs over a rough peg, and 
 
 a rough particle of -th the weight of the tube rests within 
 
 it ; find the highest position of equilibrium of the particle. 
 
 If tan ^ be the coefficient of friction between the particle 
 
 and the tube, shew that the tube will be on the point of 
 
 slipping over the peg, provided the coefficient of friction 
 
 ■1 1 T sinrf) 
 
 between the tube and peg be = ■ , ., . ., , • 
 
 n/(m + 1) -sm'<^ 
 
 25. Two weights P, Q of similar material, rest on a 
 double rough inclined plane, and are connected by a fine 
 
280 PEOBLEMS. 
 
 string passing over the common vertex : § is on the point of 
 motion down the plane — shew that the weight which may be 
 added to P without producing motion is 
 p sin 2^ sin (g + /3) 
 sin (/3 — ^) sin (a — ^) ' 
 a, 13 being the angles of inclination of the planes, and tan (j> 
 the coeflScient of friction. 
 
 26. A uniform rod rests with one extremity against a 
 rough vertical wall (/* = o) > *^® other extremity being sup- 
 ported by a string three times the length of the rod, at- 
 tached to a point in the wall ; shew that the angle the string 
 makes with the wall in the limiting position of ecjuilibrium is 
 
 tan r- or tan - . 
 27 3 
 
 ******* 
 
 27. Two equal rough balls lie in contact on a rough 
 horizontal table, and another equal ball is placed upon them 
 so that the centres of the three are in a vertical plane ; find 
 the coefficient of friction between the upper and lower balls fi, 
 and between the lower balls and the table /*', when the system 
 is on the point of motion. 
 
 Sesult. ii=3i/=2-iJ¥. 
 
 28. A rectangular table stands on a rough inclined plane, 
 and has two sides horizontal ; if the coefficient of friction of 
 the lowest feet be fi, and that of the two others be /*', find the 
 inclination (a) of the plane when the table is on the point of 
 sliding. 
 
 Remit. If the centre of gravity of the table be at a distance c from the 
 plane, and 2a be the distance between the upper and lower feet, then 
 
 tana= >'+^)" 
 
 2a+(fi'-,ty 
 
FRICTION. CHAP. III. 281 
 
 29. A straight uniform beam is placed upon two rough 
 planes, whose inclinations to the horizon are a and a', and the 
 coeflScients of friction tan\ and tanV; shew that if 6 be the 
 limiting value of the angle of inclination of the beam to the 
 horizon at which it will rest, W its weight, and B, B' the 
 pressures upon the planes, 
 
 2 tan ^ = cot (a' + V) - cot (a - \), 
 
 B B' W 
 
 and 
 
 cos\sin(a' + \') cos\'sin(a — X) sin(a— X + a'+\') 
 
 30. One end of a beam can turn in every direction about 
 a fixed point. The other rests upon the upper surface of a 
 rough plane (coefficient of friction fi), which is inclined to the 
 horizon at an angle a. If /3 be the angle which the beam 
 makes with the plane, prove that the beam will rest in any 
 
 position if tan a be not > -^^_^_^ . 
 
 31. Find the minimum eccentricity (e) of an ellipse 
 capable of resting in equilibrium on a perfectly rough in- 
 clined plane. 
 
 MesvM. If a = inclination of plane, we must have 
 e' not < 2 tan a (sec o - tan a). 
 
 32. A rod of uniform thickness is placed within a rough 
 hollow sphere, in a vertical plane passing through the centre ; 
 shew that if 6 be the inclination of the rod to the horizon, when 
 bordering upon motion, 2a the angle subtended by it at the 
 centre of the sphere, and tan j3 the coefficient of friction, then 
 
 . sin /S cos /8 
 
 tan a = -. — —57 ^ . 
 
 cos (a + p) cos (a — p] 
 
 33. A smooth sphere BCD rests against a smooth ver- 
 tical plane CH, and is propped up by a beam AB whose 
 
282 PEOBLEMS. 
 
 extremity A rests on the rougli horizontal plane EA, the 
 weights of the sphere and beam being equal. Shew that if 
 A be on the point of sliding, the angle which the beam makes 
 
 with the horizon is tan"' (— j , ju. being the coefficient of fric- 
 tion between the beam and plane. 
 
 34. Two bodies of the same weight rest upon two equally 
 rough inclined planes, being connected by a string passing 
 over the common vertex of the planes, the vertical plane 
 which contains the two bodies being at right angles to each 
 inclined plane : — if they be bordering on motion, shew that 
 the coefficient of friction is equal to the tangent of half the 
 difference of the angles of inclination of the planes to the 
 horizon. 
 
 35. A smooth sphere of radius a rests upon two parallel 
 rods, which themselves are supported upon two fixed hori- 
 zontal rods also parallel, and at right angles to the former. 
 If tan X be the coefficient of friction, and the weight of one of 
 the moveable rods be = ^ sec 1^. weight of sphere, then the dis- 
 tance between the two moveable rods in the position of rest 
 
 = 2a sin 2\. 
 
 36. A rough elliptical ring hangs across a horizontal 
 rod : shew that it will balance on it with any point in contact 
 
 if 
 
 li>- 
 
 ^Jl-e' 
 
 37. A uniform rod passes over one rough peg and under 
 another — (friction = fi). The pegs are distant h feet apart and 
 the line joining them makes an ^ ;8 with the horizon. Shew 
 that equilibrium is not possible unless the length of the rod be 
 
 >h 
 
 i + 'i^}>,. 
 
FRICTION. CHAP. III. 283 
 
 38. A rod of length a turns freely about a point which 
 is at a vertical distance c above a rough inclined plane ; the 
 lower end of the rod rests upon the plane : shew that if in its 
 position of equilibrium the vertical plane through the rod 
 cuts the inclined plane in a horizontal line, then 
 
 /. = tana.y^ 
 
 39. A rod rests in a state bordering on motion, with one 
 end fixed at a hinge and the other resting against a rough 
 vertical wall. Prove that the pressure on the hinge is to 
 the w^eight of the rod as 
 
 Vl + /4' + 4cot''a : 2 V/i,'' + cot' a, 
 
 fi being the coefficient of friction, and a the angle between the 
 rod and the wall. 
 
 FOECES NOT IN ONE PLANE. CHAPTER IV. 
 
 1. If a uniform heavy .triangle is suspended from a fixed 
 point by strings attached to the angles, the tension of each 
 string is proportional to. its length. 
 
 2. If forces act along the sides AB, A 0, BC of a triangle, 
 respectively proportional to those sides — find the line of action 
 of their resultant. 
 
 3. The line joining the hinges of a gate whose weight 
 is W is inclined at an angle a to the vertical ; shew that the 
 moment of the couple which will hold the gate in a position 
 ineliiied at a,n^^ to its position of equilibrium is proportional 
 to sin a sin /8. 
 
284 PROBLEMS. 
 
 4. A straight rod without weight is placed between two 
 pegs and forces P and Q act at its extremities in parallel 
 directions, inclined to the rod ; required the conditions under 
 which the rod will be at rest and the pressures on the pegs. 
 
 5. ABOD is a square, and forces P, 2P act along AB, 
 jBC respectively, forces 4P, 2P along AD and BC, — find the 
 locus of the points, any one of which being fixed equilibrium 
 would exist, and the pressure on such a point. 
 
 6. A string fastened at a point A supports a weight P by 
 passing under a rough handle of any form, the loose end being 
 held so that the parts on each side of the handle are parallel ; 
 find the least force which will prevent the weight from falling, 
 and the greatest which will not draw it up. 
 
 7. A heavy uniform beam has its extremities attached 
 to a string which passes round the arc of a rough vertical 
 circle ; if in the limiting position of equilibrium the beam be 
 inclined at an z 60° to the vertical, and the portion of string 
 in contact with the circle cover an arc of 270°, shew that the 
 
 J^ 
 3ot' 
 
 coefficient of friction is = -^^^ loge 3. 
 
 8. A heavy particle is attached to an endless string 
 which passes round a rough circular cylinder in a vertical 
 plane perpendicular to its axis. If in the limiting position of 
 equilibrium the string in contact with the cylinder covers an 
 arc of 270°, shew that the inclination to the horizon of the 
 two portions of the string adjacent to the particle are 
 
 tan"' e~2" and tan"' e~ "s" . 
 
 9. An unstretched elastic string just surrounds a fixed 
 square, two of whose sides are vertical, an equal square being 
 
FORCES NOT IN ONE PLANE. CHAP lY 285 
 
 introduced in the same plane as the former, and between it 
 and the lower portion of the string, just rests without touch- 
 ing it. The lower square is now turned ahout a vertical axis 
 through an i. ir, so that the string crosses between the squares ; 
 shew that the acute ^ 8 included in the position of equili- 
 brium by the two portions of the string between the squares 
 is given by the equation 
 
 sm - + 
 
 I)- 
 
 10. The ends of an elastic string without weight are 
 fastened to two points A, B, which are in the same horizontal 
 line, at a distance equal to the unstretched length of the 
 string. A weight equal to the modulus of elasticity is at- 
 tached to any point C of the string. If AD, BD be drawn 
 at right angles to AC, BC, prove that 
 
 AC BC 
 
 AB+AB^BA + BB 
 
 = 1. 
 
 11. A number of unequal weights are attached to an 
 endless string which is slung over a rough horizontal cylinder 
 so that all the weights hang free from the cylinder. Shew 
 that in the limiting positions of equilibrium the vertical 
 through the centre of gravity of the Weights divides the line 
 joining the points where the string leaves the cylinder in the 
 ratio 1 : e**", where a is the circular measure of the part of 
 the cylinder free from the string. 
 
 If the cylinder be smooth the centre of gravity of the 
 weights is vertically below the centre of the cylinder. 
 
 12. An elastic string whose natural length = c passes 
 round three rough pegs A, B, C, which form an equilateral tri- 
 angle whose side = a. The natural length of the part AB of 
 
286 PROBLEMS. 
 
 the string = c — a, and it is on the point of slipping both at 
 A and B; shew that the coefficient of friction 
 3 , /2a - cS 
 
 13, A string passes over a rough puUy (rad. = a) having 
 a concentric circular hole of radius b supported by a rough 
 axle. If the equilibrium be limiting for both, shew that 
 
 „ (1 + u°) a" — fj^b" cos a 
 
 where a is the angle of contact. 
 
 14. Three equal smooth spheres each weighing W, rest 
 within a hollow sphere of n times their radius : shew that the 
 pressure between any two of the small spheres 
 
 2W 
 
 jA{3n'-Qn-l) 
 
 15. An elastic band whose unstretched length is 2a is 
 placed round four rough pegs A, B, C, D, which- constitute 
 the angular points of a square whose side is a : if it be takeij 
 hold of at a point P between A and B and pulled in direction 
 AB, shew that it will begin to slip round A and B at the 
 
 same time, if ^P=- -, fi being the coefficient of friction. 
 
 1 + ei" 
 
 CENTRE OF GEAVITT. CHAPTER V. 
 
 1. ABGD is any plane quadrilateral figure, and a, b, c, d 
 are respectively the centres of gravity of the triangles BCD, 
 CD A, DAB, ABG; shew that the quadrilateral abed is similar 
 to ABCD. 
 
CENTRE OF GRAVITY. CHAP. V. .287 
 
 2. A triangular lamina, of -which the sides are a, b, c 
 cannot rest on its side c on a horizontal plane if c be 
 
 ■V ~3~' 
 
 3. At each of ra — 1 of the angular points of a regular 
 polygon of n sides a particle is placed, the particles being 
 equal. Shew that the distance of their centre of gravity from 
 
 the centre of the circle circumscribing the polygon is , 
 
 r being the radius of the circle. 
 
 4. From an isosceles triangular lamina ABC, of which 
 the sides AB, BG zx& equal, an isosceles portion APG\s, cut 
 away, AP, PO being equal ; (i) find G the centre of gravity 
 of the remainder. Also (ii) find the condition that it may 
 rest in neutral equilibrium when supported at the point P. 
 
 Mesvlt. Draw SPD perpendicular to AO, then (? is in this line, and 
 (i) £G=HSP+£D)—iii) BD = 2.BP. 
 
 5. Find the locus of the centres of gravity of all triangles 
 inscribed in a circle, the vertex being fixed, and the base of a 
 given length. 
 
 Result. A circle. 
 
 6. A triangular lamina ABG having a right angle at C 
 is suspended from the angle A, and the side A C makes an ^ a 
 with the horizon ; it is then suspended from B, and the side 
 BC makes an i; j8 with the horizon ; shew that 
 
 BC\ tana = AG\ta,n^. 
 
 7. If tlie sides of a triangle be taken, two and two, to 
 represent forces, acting in each case from the angle^ made by 
 
288 PKOBLEMS. 
 
 the sides, — ^prove that there is one point about which each of 
 the three pairs will balance, and find the point. 
 Remit. The point ia the centre of gravity of the triangle. 
 
 8. If the centre of gravity of a triangular pyramid be 
 the common vertex of four pyramids whose bases are the faces 
 of the original pyramid severally, shew that these four pyra- 
 mids are of equal volume. 
 
 9. A square is divided into four equal triangles by draw- 
 ing its diagonals which intersett in 0; if one triangle be 
 removed, find the centre of gravity O of the figure formed by 
 the three remaining triangles. 
 
 Bemlt. 0G= j . side of square, 
 
 10. Five pieces of a uniform chain are hung at equidistant 
 points along a rigid rod without weight, and their lower ends 
 are in a straight line passing through one end ( 0) of the rod ; 
 find the centre of gravity of the system. 
 
 Also, shew that if the system balance about a point of the 
 rod in one position it will balance about it in any position. 
 
 Sesult. If OC be drawn to C the middle point of the longest piece of chain, 
 
 the centre of gravity is in 00 and 00= ^ . 00, — the distance from to the 
 
 first piece of chain being the same as the distance between any two adjacent 
 pieces. 
 
 11. A-B, BG are two rods freely jointed at B, A is fixed ; 
 find at what point in BG a, prop must be placed so as to sup- 
 port them in a horizontal position. 
 
 12. A triangle rests in a fixed hemispherical bowl, shew 
 that the pressures at its three angular points are all equal. 
 
CENTEE OF GRAVITY. CHAP. V. " 289, 
 
 13. A straight uniform wire ABO is bent at B so that 
 the ■cABG= a, and it is then suspended by a string from the 
 point A : shew that it will rest with BG horizontal, if 
 
 BC = {AB' + 2AB. BC) cos a. 
 
 14. Explain why in ascending a hill, we appear to lean 
 forwards ; in descending, to lean backwards. 
 
 15. Why does a person rising from a chair bend his body 
 forward and his leg backward? 
 
 16. What is the use of a rope-dancer's pole ? 
 
 17. A cone whose height is equal to four times the radius 
 of its base is hung from a point in the circumference of its 
 base ; find the position in which it will rest. 
 
 Resnlt. The base and axis are equally inclined to the vertical. 
 
 18. Of what dimensions must a right cone be, in order 
 that, when the greatest sphere possible has been cut out of it, 
 the centre of gravity of the remainder may coincide with that 
 of the cone ? 
 
 Mesult. The diameter of the base : altitude of cone = 1 : a/2, 
 
 19. A smooth body in the form of a sphere is divided 
 into hemispheres, and placed with the plane of division ver- 
 tical upon a smooth horizontal plane : a string loaded at its 
 extremities with two equal weights hangs upon the sphere, 
 passing over its highest point and cutting the plane of division 
 at right angles ; find the least weight P which will preserve 
 the equilibrium. 
 
 3 
 Besidt. P—rrz weight of sphere, 
 lo 
 
 20. A weight of given magnitude moves along the cir- 
 cumference of a circle in which are fixed also two other 
 
 P. M. 19 
 
290 ■ PROBLEMS. 
 
 weights ; prove that the locus of the centre of gravity of the 
 three weights is a circle. 
 
 If the immovable weights be varied in magnitude, their 
 sum being constant, prove that the corresponding circular 
 loci intercept equal portions of the chord joining the immov- 
 able weights. 
 
 21. The three corners of a triangle are kept on a circle 
 by three rings capable of sliding along the circle, and the 
 circle is inclined to the horizon at a given angle; find the 
 positions of equilibrium. 
 
 22. If the lengths of the sides of a polygon be inversely 
 proportional to the perpendiculars let fall upon them from a 
 point 0, within the polygon, — and if G, O' be the centres of 
 gravity, respectively, of the polygon, and of a series of equal 
 heavy particles placed at its angular points, prove that OGG' 
 will be a straight line, and that 0G='2 .GG', 
 
 23. A thin uniform wire is bent into the form of a tri- 
 angle ABC, and heavy particles of weights P, Q, B are 
 placed at the angular points; prove that if the centre of 
 gravity of the weights coincide with that of the wire 
 
 P: Q: R :: AB+AG : BG + BA : GA + CB. 
 
 24. If a, )S, 7 be the feet of the perpendiculars from 
 A, B, C upon the opposite sides of the triangle ABC; p, q, r 
 the distances of the centre of gravity of triangle 0/87 from the 
 sides a, b, c of ABC, shew that 
 
 a' cos {B - C) V cos {G,^A)~ c' cos {A-B)' 
 
 25. The portion of a right cone cut oflf by a plane will 
 only just balance on a horizontal plane with the shortest side 
 
tan e - 
 
 CENTRE or GRAVITY. CHAP. V. 29t 
 
 VA in contact: prove that the vertical through A in that 
 position divides the opposite side VB in the ratio 3 : 2. 
 
 26. Three uniform rods connected hy smooth hinges 
 form a triangle ABC : — the weights of the rods being pro- 
 portional to their lengths. If the rod AB be held in a hori- 
 zontal position with the plane of the triangle vertical, shew* 
 that the direction of the strain on the hinge at G is inclined 
 to AB at an angle d given by 
 
 A-B\_ sm{A-B) 
 I )~1+CQ&{A+B)' 
 
 27. If «!, ajj, ajg be the co-ordinates of the angular points 
 of a triangle referred to any axis, the co-ordinate of the centre 
 of gravity of the triangle is = J (a?^ + x^ + x^. 
 
 And if a!j, x^, x^, x^ be the co-ordinates of the angular 
 points of a tetrahedron, the co-ordinate of its centre of gravity 
 is=l{x^+x^ + x^ + x,). 
 
 ******* 
 
 28. A, B, G, D, E, F are six equal particles at the angles 
 of any plane hexagon, and a, b, c, d, e, f are the centres of 
 gravity respectively of ABG, BGD, GDE, BEF, EFA, and 
 FAB. Shew that the opposite sides and angles of the 
 hexagon ahcdef are equal, and that the lines joining opposite 
 angles pass through one point which is the centre of gravity 
 of the particles A, B, G, D, E, F. 
 
 29. The line which joins the middle points of any two 
 opposite edges of a triangular pyramid is bisected by the 
 centre of gravity of the pyramid. 
 
 30. From the fact that a system of heavy particles has 
 one centre of gravity only, shew that the lines joining the 
 
 19-3 
 
292 PROBLEMS. 
 
 middle points of opposite sides of any plane quadrilateral 
 bisect each other. 
 
 31. If the centre of gravity of a four-sided figure coincide 
 with one of its angular points, shew that the distances of thig 
 point and of the opposite angular point from the line joining 
 the other two apgular points are as 1 : 2. 
 
 32. A cone whose semi vertical angle is tan"' . is 
 
 V(2} 
 
 enclosed in the circumscribing spherical surface ; shew that it 
 will rest in any position. 
 
 33. Give a geometrical construction for finding the centre 
 of gravity of a plane quadrilateral area. 
 
 34. If G be the centre of gravity of a triangle ABG, 
 shew that 
 
 Z[AO' + BG' + CG') = AB' + BC + CA\ 
 
 35. Two sides AB, CD of a quadrilateral are parallel, 
 and their middle points 0, T are joined by a line OT oi 
 length c ; if AB — a, CD = h, and G be the centre of gravity 
 of the figure, shew that 
 
 c a + ib 
 
 0G = 
 
 3" a + i 
 
 36. A pack of cards is laid on a table, and each projects 
 in direction of the length of the pack beyond the one below 
 it; if each projects as far as possible, prove that the distances 
 between the extremities of saccessive cards will form a har- 
 monic progression. 
 
 37. Prove the following geometrical construction for the 
 centre of gravity of any quadrilateral. Let E be the inter- 
 section of the diagonals, and F the middle point of the line 
 
CENTRE OF GEAVITT. CHAP. V. 293 
 
 which joins their middle points ; draw the line E^ and pro- 
 duce it to (?, making FGr equal to one- third of EF; then Q 
 shall be the centre of gravity required. 
 
 38. A right cone whose axis is a, and vertical angle 
 
 2 sin"* . / (- J , is placed with its base in contact with a smooth 
 
 vertical wall, and its curved surface on a smooth horizontal 
 rod parallel to the wall ; shew that it will remain at rest if 
 
 the distance of the rod from the wall be not > a nor < - . 
 
 7 
 
 39. The weights of three particles A, B, at the angu- 
 lar points of the triangle ABG are respectively proportional 
 to the opposite sides of the triangle ; the centre of gravity of 
 the three particles coincides with the centre of the circle in- 
 scribed in the triangle. 
 
 40. A piece of uniform wire is bent into three sides of a 
 square ABCD of which the side AD Is wanting; shew that 
 if it be hung up by the two points A and B successively, the 
 angle between the two positions of BG is tere"' 18, 
 
 41. A frustum is cut from a right cone by a plane bisect- 
 ing the axis and parallel to the base. Shew that it will rest 
 with its slant side on a horizontal table if the height of the 
 cone bear to the diameter of the base a greater ratio than 
 
 VT : ^717. 
 
 42. A weight. W Is placed at on. & triangular table 
 ABC, supported in a horizontal position by three props at 
 the angular points ; shew that the portions of the weight sus- 
 tained by the props nt A, B, G are proportional to the areaS 
 of the triangles BOG, A OG, A OB. 
 
294 PROBLEMS. 
 
 43. A right-angled triangle is suspended successively 
 from its acute angles, and when at rest, the side opposite the 
 point of suspension in each case makes angles 6, ^ with the 
 vertical, — shew that tan 5 tan ^ = 4. 
 
 44. Through the angles of a triangular board lines are 
 drawn to the opposite sides, each dividing the triangle into 
 two equal parts. Shew that the area of the figure formed hy 
 joining the centres of gravity of these parts is | of the area 
 of the triangle. 
 
 45. A heavy square board of uniform thickness is sus- 
 pended freely by one corner : and at each end of the diagonal 
 which does not pass through that corner a weight is sus- 
 pended, — shew that the inclination of that diagonal to the 
 
 horizon is = tan"' i p , /i , Ty ) j — where P, Q are the 
 weights and W the weight of the board. 
 
 46. Parallel forces act at the angles A,B, C oi a triangle, 
 and are respectively proportional to the sides a, h, c, — shew 
 that their resultant acts at the centre of the inscribed circle. 
 
 47. Prove the following rule for finding the centre of 
 gravity of any quadrilateral lamina ABGD. a, c are the per- 
 pendicular distances of A and C from BD. Take i^ in J. C 
 such that FA : FG :: c : a. Join F with E the middle 
 point of BD and take OE=^EF. G is the centre of gravity 
 required. 
 
 it it If ■* It ^ ii 
 
 48. If the vertical angle of a right cone of circular base 
 be > sin"' \, the upper frustum cut off by any plane will be 
 supported with its base on a horizontal plane. 
 
CENTRE OF GRAVITY. CHAP. V. 295 
 
 If the vertical angle be < sin~* ^, determine the limits for 
 the inclination of the cutting plane to the axis that the frustum 
 may stand. 
 
 49. A heavy right cone rests with its base on a fixed 
 rough sphere of given radius, determine the greatest height of 
 the cone compatible with stability. 
 
 50. Find the centre of gravity of an Isosceles triangle, 
 out of which an inscribed square has been cut. 
 
 Hesult. If B=0 in the triangle ABC and AD he drawn from ^ perpen- 
 dicular to the base, G the centre of gravity required lies in AD, and if 
 
 /.A = 2a 
 
 ■ ^2 1 +6tan''a + 8tan3a 
 
 3 (1+2 tan a) (1+4 tan^ a)' 
 
 51. A triangular prism, each side of whose base = a, 
 
 rests symmetrically between two smooth parallel horizontal 
 
 bars at a distance = 2c from each other ; if the prism be 
 
 divided into two equal parts by a vertical plane which 
 
 bisects the lowest angle of the prism, the parts will remain 
 
 5 1 
 
 in equilibrium, provided c be < — - a and > — a. 
 
 12 24 
 
 52. A cube has two fequal portions cut off by planes 
 passing through a diagonal of one of its faces and two 
 comers of the opposite face. If it be suspended freely from 
 one of the extremities of the diagonal, shew that its two 
 
 remaining edges will be inclined at tan"' — — - to the vertical. 
 
 53. Two pieces of flexible chain of different weights 
 but of equal lengths are fastened together so as to have a 
 common extremity. They are then laid over a smooth verti- 
 cal circle resting wholly in contact with it. Find the posi- 
 tion of equilibrium. 
 
296 PEOBLEMS. ' 
 
 54. A piece of uniform heavy wire is formed into a' tri- 
 angle ABO, and the middle points of the sides are joined by- 
 pieces of wire of the same thickness. If the framework so 
 formed he hung up from the z A, shew that AB, A G make 
 with the vertical angles 0, <p such that 
 
 sin 6 5c (« + c) + 25c 
 sin^ ~ 56 {a + b) + 2hc ' 
 
 55. The centres of gravity of the area and perimeter of 
 a polygon circumscribed about a circle, lie on a diameter; 
 and their distances from the centre are as 2 : 3. 
 
 56. If ABG be an isosceles triangle having a right angle 
 at C, and I), E be the middle points oi AC, AB respectively, 
 prove that a perpendicular from E upon BD will pass through 
 the centre of gravity of the triangle BDG. 
 
 57. From a given rectangle ABGD cut off a triangle CDO 
 (the point being in AD) so that when the figure ABCO is 
 suspended from the sides A 0, BG may be horizontal. 
 
 Jlesult. AO : AD= ^/S-l : 2. 
 
 58. A uniform beam of thickness 2b rests symmetrically 
 on a perfectly rough horizontal cylinder of radius a ; — shew- 
 that the equilibrium of the beam will be stable or unstable 
 according as b is less or greater than a. 
 
 59. A uniform wire is bent into the form of three sides 
 AB, BO, CD of an equilateral polygon, and its centre of 
 gravity is at the intersection of A G, BD ; shew that the 
 polygon must be a regular hexagon. 
 
 60. A pyramid, the base of which is a square, and the 
 other faces equal isosceles triangles, is placed in the circum- 
 scribing spherical surface; prove that it will rest in any 
 
CENTRE OF GEAVITT. CHAP. V. 297 
 
 position, if the cosine of the vertical angle of each of the 
 triangular faces be = f . 
 
 61. Two equal heavy particles are situated at the ex- 
 tremities of the latus-rectum of a parabolic arc without weight, 
 which is placed with its vertex in contact with that of an 
 equal parabola whose axis is vertical and concavity down- 
 wards ; prove that the parabolic arc may be turned through 
 any angle without disturbing its equilibrium, provided no 
 sliding be possible between the curves, 
 
 62. Find the centre of gravity of the volume Included 
 between two similar parallelopipeds which have a common 
 angle. Also determine the limiting position of the centre of 
 gravity when the parallelopipeds approach equality. 
 
 63. The centres of two circles which touch each other 
 internally are made to approach indefinitely near to each other, — 
 find the ultimate position of the centre of gravity of the area 
 included between the circumferences of the circles. 
 
 Also find according to what power of the distance from 
 a fixed point in the circumference the density of a circular 
 wire must vary, that its centre of gravity may coincide with 
 that of the above figure. 
 
 64. The centres of gravity of the area and perimeter of a 
 plane triangle lie in a line which passes through the centre of 
 the inscribed circle, at distances from it which are as 2 :. 3. 
 
 65. If n lines drawn from a point represent in magnitude 
 and direction a system of forces acting at that point, shew 
 that the resultant of the system of forces will be represented 
 in magnitude and direction by n times the line drawn from 
 that point to the centre of gravity of n equal partieks placed 
 at the extremities of the lines. \ 
 
298 PEOBLEMS. 
 
 66. The centre of the circumscribing circle of any triangle 
 is the centre of gravity of four equal particles placed at the 
 centres of the inscribed and escribed circles, 
 
 67. If a portion of a parabola bounded by the latus 
 rectum {L) be placed with its vertex on that of a given 
 cycloid, the convexities of the two curves being turned in 
 opposite directions, the equilibrium will be neutral if ZL = 28a, 
 where a equals diameter of generating circle of the cycloid. 
 
 68. An elliptic cylinder, whose semiaxes are a, b, rests 
 between two smooth inclined planes at right angles to one 
 another, prove that there will be three positions of equi- 
 librium if the inclinations of the planes to the horizon be 
 
 > tan ' - . 
 a 
 
 69. A plane quadrilateral ABCD is bisected by the 
 diagonal AC, and the other diagonal divides AC into two 
 parts in the ratio p '• g^; shew that the centre of gravity of 
 the quadrilateral lies in ^ C and divides it into two parts in 
 the ratio 2p + g :p + 2q. 
 
 70. If a right-angled triangular lamina ABC be sus- 
 pended from a point I) in its hypotenuse AB, — prove that in 
 the position of equilibrium, AB will be horizontal if 
 
 AD : BB :: AB'+AC : AB'+BC\ 
 
 71. If <? be the centre of gravity of a system of particles, — 
 D, A the distances of any one of them m from G, and (any 
 other point), — shew that 
 
 S {mB') = S (mA=) - 2 {m) GO'. 
 
 72. AB, BC are rods having a joint at B, A being a 
 fixed hinge ; find the position in which the system will rest 
 
MACHINES. CHAP. VI. 299 
 
 when a string from A is attached to a ring sliding on BG, 
 supposed smooth. Find also the tension of the string. 
 
 73. In a triangular pyramid ABOD if a, h, c be the sides 
 of the triangle ABC,, and a, /3, y the edges meeting in B, 
 shew that if G be the centre of gravity of the pyramid 
 
 BG = i{3 {u'+^+y') - {a'+V + c')]i. 
 
 74. If each particle of a system be multiplied by the 
 square of its distance from an assumed point 0, the sum of 
 these products will be least when coincides with the centre 
 of gravity of the system of particles. 
 
 MACHINES. CHAPTER VI. 
 
 1. The arms of a balance are equal in length, but one 
 scale is loaded ; find the true weight of the body in terms of its 
 apparent weights when suspended at each end in succession. 
 
 Beiult. The true weight = semi-sum of the apparent weights. 
 
 2. Two men A, B of the same height bear a weight 
 hung on a pole which rests on their shoulders ; where must 
 the weight be placed in order that A may support n times as 
 much as 5? 
 
 Seivlt. The distance of the weight from B must =ra, times its distance 
 from A. 
 
 3. A uniform steel rod AB having a constant weight 
 P, and a scale-pan of weight AP, suspended at B and A 
 respectively, is used as a balance by moving the rod back- 
 
300 PROBLEMS. 
 
 ■wards and forwards upon the fulcrum O on which the whole 
 rests. Shew that the beam must be graduated by the formula 
 
 , ^' 
 
 n + k + k +1 
 
 the weight of the rod being h'P, and n being each of the 
 natural numbers 1 . 2.3... taken in succession. 
 
 4. If thepifcA of a screw be - , tan ^ the coefficient of 
 
 friction, P the least force which will prevent the weight from 
 descending, P' the greatest which can be applied without 
 its rising, then 
 
 pr:j:-p=sm2^. 
 
 5. Weights of 3oz. and \\h. balance on a straight 
 lever of which the longer arm is 2 feet ; find the length of the 
 shorter arm. 
 
 MemLt. 9 inehes. 
 
 6. In .any system of puUies in which a separate string 
 passes over each pully and the strings are parallel, prove that, 
 if the tensions of the strings increase in geometric progression, 
 so do the weights of the pullies. 
 
 7. Two weights P, Q are connected by a string PAQ 
 passing over a pully A, — P hangs vertically and Q rests on a 
 rough inclined plane (a) and X is the angle of friction : — if 
 the greatest and least angles which A Q can make with the 
 plane be e, \, shew that 
 
 tan' — -— = tanX . cot a. 
 
MACHINES. CHAP. VI. 301 
 
 8. If P support Won & rough inclined plane (a), P act- 
 ing in a principal plane and at an angle e with the plane, 
 and if P may have any magnitude intermediate to F, P" 
 without producing motion and the plane be but slightly 
 
 1 , ,, , P' ~ P" sin a cos e , 
 
 rough, shew that /* = ,^;— — r, . -^-Ji—— nearly. 
 Jr" + Jr sin (a+ej "^ 
 
 In what cage will this be the exact value of ytt? 
 
 9. The length of- the shorter arm of a common steelyard 
 = 4 inches : the removal of P through ^ {nch indicates an in- 
 crease of 2 oz. in the weight W: — and the notch correspond- 
 ing to a weight of 4 lbs. is 3 inches from the fulcrum : — 
 determine the moment of the beam. 
 
 10. If the same body be weighed successively at the two 
 ends of a false balance whose arms are of unequal length, its 
 true weight is the square root of the product of the apparent 
 weights. 
 
 11. If a man sitting in one scale of a weighing-machine 
 press with a stick against any point of the beam between the 
 point from which the scale is suspended and the fulcrum, he 
 will appear to weigh more than before. 
 
 12. Explain how a man by walking slowly up the surface 
 of a large rough sphere may make it roll up an inclined plane 
 or along a horizontal plane in any dia'ection. 
 
 13. If a tradesman's balance have unequal arms, a, h, 
 and he weighs goods alternately fropa one scale and the other, 
 does he gain or lose by his balance not being true ? and how 
 much ? 
 
 BesuU. His loss : a/pparent weight which he dispenses :: {a-Vf : 2a5, 
 
302 PROBLEMS. 
 
 14. The sensibility of a Danish steelyard at any point 
 varies as the square of the distance of the point from the end 
 where Wis suspended. 
 
 15. If a uniform wire he bent into the form of a triangle, 
 and at the middle points of the sides there be placed three 
 beads whose weights are proportional to the sides on which 
 they are ; prove that when the beads are moved with equal 
 velocities in the same direction along the sides there will be 
 no change in the position of the centre of gravity of the 
 whole system. 
 
 16. If two weights support each other on inclined planes 
 by means of a string passing over the common vertex of the 
 planes, and the system is set in motion, the centre of gravity 
 of the weights moves in a horizontal line. 
 
 17. When F supports W on a, rough inclined plane, and 
 B is the pressure on the plane, explain the result when 
 
 e+^is>90''. (Art. 110.) 
 
 18. In the system of pullies where each string is at- 
 tached to the weight, if one of the strings be nailed to the 
 block through which it passes, shew that the power may be 
 increased up to a certain limit without producing motion. 
 If there be three pullies, and the action of the middle one be 
 checked in the manner described, find the tension of each 
 string for given values of P and W, 
 
 19. In a wheel and axle, if the axis about which the 
 machine turns coincide with that of the axle but not with 
 the axis of the wheel, find the greatest and least ratios of the 
 power and weight necessary for equilibrium, neglecting the 
 weight of the machine. 
 
MACHINES. CHAP, YI. 303 
 
 20. Why is it easier to move a heavy body when placed 
 upon rollers than to draw it upon a rough horizontal plane ? 
 Compare the rates of motion of the body and of the centres of 
 the rollers. 
 
 21. In the system of pullies of Article 106 — if the weight 
 of the lowest puUy be equal to the power P, of the second 
 3P, and so on, that of the highest movable puUy being 
 S'-'P— the ratio of Pto TFwill be 2 : 3"- 1. 
 
 22. In the Danish Steelyard, if a„ be the distance of the 
 fulcrum from that end of the steelyard at which the weight 
 is suspended, the weight being n lbs. prove that 
 
 1 2 1^ 
 +- = 0. 
 
 "•"+1 
 * 
 
 23. In each of the three systems of pullies, if P and 
 W receive any displacement their centre of gravity remains 
 unchanged in position. 
 
 24. If three forces P, Q, B are in equilibrium when act- 
 ing on a particle, and the particle be slightly displaced so that 
 J), q, r are the virtual velocities of P, Q, B respectively, shew 
 that Pp+ Qq + Br = 0. 
 
 Prove the principle of virtual velocities in the case of the 
 Spanish Barton. (Art. 107.) 
 
 25. In the system of pullies where each hangs by a 
 separate string, determine the relation between the radii of 
 the pullies in or'der that, if their centres be at any time in a 
 straight line, they may always continue to be so. 
 
 26. K a common steelyard be constructed with a given 
 rod, ■#hose weight is inconsiderable compared with that of 
 
304 PEOBLEMS. 
 
 the sliding weiglit, sliew that the sensibility varies inversely 
 as the sum of the sliding weight and the greatest weight 
 which can be weighed. 
 
 27. A heavy insect of weight w crawls on the lower cir- 
 cumference of the wheel of a wheel and axle, and so just raises 
 a weight 5w, the ratio of the radii of the wheel and axle being 
 10 : 1, — find the inclination to the vertical of the radius of 
 the wheel which passes through the position of the insect : — 
 shew that the insect is in a position of stable equilibrium, 
 but that if it were on the upper surface of the wheel and 
 at a point vertically above its present position, its equili- 
 brium would be unstable. 
 
 28. If a wheel and axle be similar coaxial regular prisms 
 so placed that every plane bisecting an angle of one bisects a 
 side of the other, shew that the ratio of the least to the great- 
 est power which will support a given weight is cos"- : 1, 
 where n is the number of faces of the prism. 
 
 29. If a power P balance a weight TFin a combination of 
 n movable pullies, each of weight w, shew that 
 
 W= (P+ w) (2"« - 1) _ (« + 1) (o, 
 
 the cords being parallel and each attached to the weight. 
 
 Also if the weights of the movable pullies be P, 2P, 
 3P,... the puUy whose weight is P being furthest from the 
 weight, shew that 
 
 Tr=p|«2»«-'^i|tl)+i 
 
 30. Apply the principle of Virtual Velocities to de- 
 termine the ratio of the power to the weight, when the weight 
 slides along a smooth vertical rod, and is attached by an. 
 
MACHINES. CHAP. VI. 305 
 
 inextensible string to a point in the rod, while the power acts 
 horizontally at the middle point of the string. 
 
 31. A heavy particle rests on a rough inclined plane (a), 
 being attached to a point in the plane by a string which 
 makes an angle 6 with the line of greatest slope down the 
 plane ; — ^find the tension of the string, and shew that must 
 not be > sin "' (/* cot a) : where /j, = coefficient of friction. 
 Explain this result if /t > tan a. 
 
 32. A rod AB, whose weight (p) and centre of gravity G 
 are given, is to be used as a Danish balance, the substance to 
 be weighed being suspended from B; A^, A^,...A„ the points 
 where the fulcrum is to be placed to weigh 1, 2,...« pounds 
 respectively, are marked by pins (each of a given weight w) 
 being driven in ; find a formula for the graduation. 
 
 Remit. If ^0 be the position of the fulcrum when there is no weight at B, 
 £Af,=z, AfiAi=Xi, AiA2—,x^...A,-iAr=Xr...we shall have m equations of the 
 type 
 
 rz 
 
 iCr= a:. — iCo— ... —Xr~l 
 
 r+p+noj 
 
 by giving r Buooessive integral values 1.2. Z...n, which together with the 
 equation 
 
 are sufficient to determine the n ■)- 1 quantities z, Xi, X2. ..a!i,. 
 
 33. A person suspended in a balance of which the arms 
 are equal, thrusts his centre of gravity out of the vertical by 
 means of a rod fixed to the furthest extremity of the beam 
 of the balance, the direction of the rod passing through his 
 centre of gravity ; given that the rod and the line from the 
 nearer end of the beam of the balance to his centre of gravity 
 make angles a, with the vertical, shew that his apparent 
 and true weights are in the ratio 
 
 sin (a + yS) : sin (a - /S). 
 p. M. 20 
 
306 PEOBLEMS. 
 
 MISCELLANEOUS EXAMPLES IN STATICS. 
 
 1. A 'body consisting of a cone and hemisphere having 
 the same base, is placed upon a rough horizontal tahle ; deter- 
 mine the greatest height of the cone that the equilibrium may 
 be stable. 
 
 Remit. Altitude of cone = ijz . radius of the hemisphere. 
 
 2. A solid is composed of a cylinder and hemisphere of 
 equal radius, fixed base to base ; find the ratio of the height 
 to the radius of the cylinder, that the equilibrium may be 
 neutral when the spherical surface rests on a horizontal plane. 
 
 Result. Altitude of cylinder =-p radius. 
 
 3. When a man stands on a hill, how is he inclined to 
 the horizon and to the hill ? 
 
 4. Two forces F and F' acting in the diagonals of a 
 parallelogram, keep it at rest in such a position that one of 
 its edges is horizontal ; shew that 
 
 F sec a' = -F* sec a = 1^ cosec (a + a'), 
 where W is the weight of the parallelogram, a and a! the 
 angles between its diagonals and the horizontal side. 
 
 5. A cylinder rests with the centre of its base in contact 
 with the highest point of a fixed sphere, and four times the 
 altitude of the cylinder is equal to a great circle of the sphere ; 
 supposing the surfaces in contact to be rough enough to pre- 
 vent sliding in all cases, shew that the cylinder may be made 
 to rock through an angle of 90°, but not more, without falling 
 off the sphere. The base of the cylinder being supposed to be 
 sufficiently large. 
 
MISCELLANEOUS EXAMPLES IN STATICS. 307 
 
 6. If three parallel forces acting at the angular points 
 A, B, O of a triangle are respectively proportional to the op- 
 posite sides a,h, c; prove that the distance of the centre of 
 parallel forces from A 
 
 ibc A 
 
 — — , — cos — . 
 a+h+c 2 
 
 7. Two equal spheres placed in a paraboloid with its 
 axis vertical touch one another at the focus. If W be the 
 weight of a sphere, B, E the pressures upon it, prove that 
 
 W^:B.B'::Z:2. 
 
 8. Three equal cylindrical rods are placed symmetrically 
 round a fourth one of the same radius, and the bundle is then 
 surrounded by two equal elastic bands at equal distances from 
 the two ends ; if each band when unstretched would just pass 
 round one rod, and a weight of lib. would just stretch one to 
 twice its natural length, shew that it would require a force of 
 9 lbs. to extract the middle rod, the coefficient of friction 
 
 being equal to — . 
 
 9. ABCD. . . is a string without weight suspended from 
 two points A, F va. a, horizontal line; and given weights W^, 
 W^, TFj-.-are hung from the knots B, 0, D...; determine 
 the proportion of these weights when the string hangs in a 
 given form. (N.B. This is called a funicular polygon.) 
 
 If the weights be all equal, shew that the co-tangents of 
 the angles which successive portions of the string make with 
 the vertical are in arithmetic progression. 
 
 10. Two strings of the same length have each of their 
 ends fixed at each of two points in the same horizontal plane. 
 A smooth sphere of radius r and weight W is supported upon 
 
 20—2 
 
308 PKOBLEMS. 
 
 tbem at tbe same distance from each of the given points. 
 
 If the plane in which either string lies makes an angle a with 
 
 Wa 
 the horizon, prove that the tension of each is = -^r— cosec a ; 
 
 a being the distance between the points. 
 
 11. Strings are fixed to any number of points A, JB, C... 
 in space, and are pulled towards a point P with forces propor- 
 tional to PA, PB, P ; shew that wherever the point P be 
 situated the resultant of these forces will always pass through 
 a fixed point. 
 
 12. Two equal weights P, P are attached to the ends of 
 two strings which pass over the same smooth peg, and have 
 their other extremities attached to the ends of a beam AB 
 (weight W) which rests thus suspended ; shew that the incli- 
 nation of the beam to the horizon is 
 
 = tan"' I — -y tan a ] : 
 \a+b J 
 
 a, h being the distances of the centre of gravity of the beam 
 
 from its ends, and sina = — p. 
 
 13. A particle is placed in the middle point of a hori- 
 zontal, equilateral, and triangular board, and is kept in equi- 
 librium by three equal weights, which act by means of strings 
 passing through the angular points. When the particle is 
 moved in direction of one of the angular points, find the force 
 tending to restore it to its position. 
 
 If the force be half of the weight, the inclination -of the 
 
 strings will be = cos"' (— t) • 
 
MISCELLANEOUS EXAMPLES IN STATICS. 309 
 
 14. A cylinder — ^length h, diameter c — open at the top, 
 stands on a horizontal plane, and a uniform rod — ^length 2a — 
 rests partly within the cylinder, and in contact with it at its 
 upper and lower edges ; supposing the weight of the cylinder 
 to be n times that of the rod, find the length of the rod when 
 the cylinder is on the point of falling over. 
 
 Eesvlt. 2(1 = (m+2) Jbi' + c'. 
 
 15. A uniform bent lever whose arms are at right angles 
 to each other, is capable of being enclosed in the interior of 
 a smooth spherical surface, — determine the position of equi- 
 librium. 
 
 Result. The arma of the lever will be equally inclined to the vertical. 
 
 16. If c be the length of the axis of a frustum of a 
 pyramid, — a, h homologous sides of its larger and smaller 
 ends, the distance of the centre of gravity from the end a — 
 measured along c — is 
 
 ~4" d' + ab + F' 
 AVhat does this become (i) when a = &, (ii) when J = ? 
 
 17. If a triangle be supported in a horizontal position 
 by vertical threads fastened to its angular points, each of 
 which can just bear an additional tension of 1 lb., determine 
 within what portion of the area a weight less than 3 lbs. may 
 be placed without destroying the equilibrium. 
 
 18. A square — ^whose side = 2a — is placed with its plane 
 vertical between two smooth pegs which are in the same 
 horizontal line at a distance c ; shew that it will be in equi- 
 librium when the inclination of one of its edges to the horizon 
 
 . . .d'—c" IT 
 
 = A sm-' — J— . or = - . 
 ^ ' c' 4 
 
310 PEOBLEMS. 
 
 19. A sphere rests upon a string fastened at its extre- 
 mities to two fixed points ; shew that if the arc of contact of 
 the string and sphere be not <2tan~'f|, the sphere may 
 be divided into two equal portions by means of a vertical 
 plane without disturbing the equilibrium. 
 
 N.B. The centre of gravity of a half sphere, is at a distance from the 
 centre of the spherical surface equal to f of the radius. 
 
 20. A polygon of an even number of sides is formed by 
 a number of rods which are connected by free joints at their 
 extremities, and is kept in equilibrium by forces applied 
 perpendicularly to the rods at their middle points — shew that 
 the sums of the alternate angles are equal. 
 
 If the polygon be of an odd number (2w + l) of sides, 
 and Kj, aj...a2„+, be the angles, — shew that the direction of the 
 strain at a^ on the side adjacent to Kj, a^n+i is inclined to that 
 
 ^ l» ^ "of 
 
 side at an angle whose complement is " "^' - — — '^, the 
 forces being all supposed to tend inwards. 
 
 21. An endless elastic string (without weight) when 
 unstretched, just passes round two pegs in a horizontal plane : 
 two weights W, W are hung upon it in such a manner that 
 the string forms two festoons, the angles in these being 20, 
 2^ respectively : shew that if \ be the modulus of elasticity, 
 
 then 
 
 W W 
 
 cosec d + cosec ^ — 2 = — - sec = — - sec rf>. 
 
 A A. 
 
 22. Three equal rods connected by two free joints are 
 attached by similar joints to two points in the same horizontal 
 plane. If the rod next to one of these joints makes an ^ a 
 -with the horizon — ^and the reaction on the joint at its lower 
 end an / 6, — then tan = J tan a. 
 
MISCELLANEOUS EXAMPLES IN STATICS. 311 
 
 23. A heavy equilateral triangle hung up on a smooth 
 peg by a string, the ends of which are attached to two of its 
 angular points, rests, with one of its sides vertical — shew that 
 the length of the string is double the altitude of the tri- 
 angle. 
 
 24. A fine string A GBP tied to the end ^ of a uniform 
 rod AB of weight W, passes through a fixed ring at C, and 
 also through a ring at the end B of the rod, the free end of 
 the string supporting a weight P : if the system be in equi- 
 librium, prove that AC : BG :: 2P + W : W. ' 
 
 25. A vertical cylinder is cut into parts by a plane in- 
 clined at an ^ a to the axis, and the parts are held together 
 by a string passing in a horizontal plane round the cylinder, 
 find the tension of the string, and shew how it varies for 
 different positions of the string : — the common surface of the 
 two parts being smooth. 
 
 26. AB, BG are two equal uniform beams united by a 
 free joint at B, and hanging freely from a peg at A to which 
 is attached a string passing to G; — prove that the action at 
 the joint is to the weight of each beam as 
 
 V4-3cos''C : 2 V4 - 3 sin" G. 
 
 27. A picture frame is supported by one cord, which 
 passes* over a smooth peg and through two smooth rings, 
 symmetrically situated at the back of the frame : the cord is 
 weightless and elastic, and when unstretched, it just reaches 
 through the rings: — e being the modulus of elasticity, and w 
 the weight of the frame. Shew that the vertical / (2a) of 
 the triangle formed by the string is determined by the equa- 
 tion e (1 — sin a) =w tan a. 
 
312 PROBLEMS. 
 
 28. Two small smooth rings of equal weight slide on a 
 smooth elliptical wire of which the axis major is vertical, and 
 are connected by a string passing over a smooth peg at the 
 upper focus: — prove that the rings will rest in whatever 
 position they may be placed. 
 
 29. A right cone is held with its base against a rough 
 vertical wall by means of a string attached to its vertex, and 
 to a point of the wall vertically above the highest point of its 
 base : — find the greatest length of the string for which eqm- 
 librium in su5h a position is possible. 
 
 30. A rectangular board whose sides are a, h, and weight 
 W, is supported in a horizontal position by vertical strings at 
 three of its angular points, — a weight 5 W being placed on the 
 board the tensions of the strings become W, 2W, 3 W; find 
 all the positions of the weight. 
 
 Compare Prob. 42, p. 293. 
 
 31. Two weights support each other on two smooth in- 
 clined planes, which have a common vertex, by means of a 
 string which passes over a smooth pully at a given height 
 vertically above the vertex ; find the position of equilibrium, 
 and, if the planes themselves be capable of motion along a 
 smooth horizontal plane, determine the horizontal force neces- 
 sary to keep them at rest. 
 
 32. Any number of forces act upon a rigid body in one 
 plane, — one point being supposed fixed, whose co-ordinates 
 X, y are given by the equations 
 
 xtX+y% Y= t{xX + yY); 
 
 xtY-ytX=t{xY-yX); 
 
 prove that the forces will keep the body at rest ; and will also 
 
MISCELLANEOUS EXAMPLES IN STATICS. 313 
 
 keep it at rest if their directions be also turned thrdugli any 
 given angle. 
 
 33. A number n of particles of equal weight w are 
 fastened to an endless inelastic thread of given length c, 
 at equal distances from each, and the necklace so formed is 
 placed on a smooth cone (2a) with its axis vertical and 
 vertex upwards ; find the tension t of the portions of thread, 
 and the distance x of each particle from the vertex of the 
 cone. 
 
 Deduce the tension 7 of a heavy string ]^ placed in the 
 same manner on the cone. 
 
 Sesult. i = — ooa o coseo ( 1 , 
 
 2 \ n J' 
 
 c fir sin a\ _ „^ cot a 
 x= — cosec ) , T= W -=. — . 
 
 34. A thin rod rests in a horizontal position between two 
 rough planes equally inclined to the horizon and whose incli- 
 nation to each other is 2a ; if /* be the coefficient of friction, 
 shew that the greatest possible inclination of the line of inter- 
 section of the planes to the horizon is tan"' . -4—- . 
 
 sma 
 
 35. The line of intersection of two smooth planes A, B 
 is horizontal; a rod OD rests first with its extremity in 
 contact with the plane A, and secondly with the extremity D 
 in contact with the same plane. If 6, <j) be the inclinations of 
 the rod to the horizon in these two positions of equilibrium, 
 prove that tan 6 + tan is invariable, whatever be the 
 length of the rod, or the position of its centre of gravity. 
 
 36. Three rods OA, OB, OC are jointed together at 
 in such a manner that they can be fixed in any position in 
 
314 PROBLEMS. 
 
 whlcli the angles they make with one another are not less 
 than right angles. The system is then placed successively 
 on each of its three points A,JB,C with the lower rod ver- 
 tical, the angle between the upper two being the greatest 
 possible. If a, /3, 7 be the values of this angle in the three 
 cases (a the least of them) prove that 
 
 cos a + cos /8 cos 7 = 0. 
 
 37. Two smooth rings are connected with a third by 
 inextensible strings without weight. The three rings slide 
 on a smooth wire bent into the form of a vertical circle. 
 Find the position of equilibrium : and prove that, if the mass 
 of each ring be multiplied by its distance from the vertical 
 diameter of the wire, the algebraical sum of the products 
 (considered of different signs when the rings are on opposite 
 sides of the diameter) will be zero. 
 
 38. A rod AB is placed in a fixed smooth hemispherical 
 bowl of radius c, so as to lean against the edge of the bowl 
 at P, with one end A within it. Find the position of equi- 
 librium. 
 
 Result. If ij> be the inclination of the rod to the horizon, it is detennined 
 
 by the equation 
 
 a 
 cos 20= — coa (b. 
 '^c 
 
 39. Three equal right cones stand on a rough horizontal 
 plane with the rims of their bases in contact with each other 
 and a heavy smooth sphere is placed between them. If the 
 vertical angle of each cone be 60°, and the coeflficient of fric- 
 tion for the surface in contact be cot 60°, shew that the greatest 
 weight of the sphere consistent with equilibrium is two- 
 thirds of the weight of each cone: and find the magnitude 
 
MISCELLANEOUS EXAMPLES IN STATICS. 315 
 
 and position of the sphere if the cones are on the point of 
 falling over. 
 
 40. A weight P being placed upon a triangular table, 
 place another given weight Q upon the table in such a posi- 
 tion that the. pressure on the three props at the angles may be 
 equal. Within what limits is the problem possible ? 
 
 Employ Prob. 42, p. 293. 
 
 41. If an even number of uniform beams of equal length 
 
 and weight rest in equilibrium in the form of an arch, and 
 
 ttj, aj...a„ be the respective angles of inclination of the first, 
 
 second... n**" beams to the horizon, counting from the top, 
 
 prove that 
 
 2w+l . 
 tano;,^j = 2^^--^.tana„. 
 
 42. ABGG, DEF are two horizontal levers without 
 weight, B, F their fulcrums ; the end D of one lever rests 
 upon the end of the other ; HK is a rod without weight 
 suspended by two equal parallel strings from the points E, O. 
 Prove that a weight F zX A will balance a weight W placed 
 any where on the rod SK, provided 
 
 EF_BG . P__^ 
 DF~ BC W~ AB' 
 
 43. Two equal particles {w, w) are connected by two 
 given strings (2c, 2c') without weight, which are placed like a 
 necklace on a smooth cone (2a) with its axis vertical and 
 vertex upwards ; find the tensions of the strings. 
 
 Jlesvlt. The tensions t, f are given by the equations 
 i f w cos a 
 
 COS <p' cos <j> sin (ir sin a) ' 
 c sin (if sin a) , ,, c' sin ( tt sin a) 
 
 *°<^ **°*'=c' + coos(,rsma)' *^ ^ " ^T^'Tos (tt sin a) " 
 
316 PROBLEMS. 
 
 44. Two particles are joined to a string, and the system 
 is in equilibrium on the convex surface of a cycloid whose 
 axis is vertical, and convexity upwards ; shew that the dis- 
 tances along the cycloid from the highest point are inversely 
 proportional to the weights. 
 
 45. A sphere of given weight rests upon three planes 
 whose equations are Ix + my + ws = ; l^x + m^y + n^z = (i; 
 l^x + m^y + Wj3 = ; the axis of z heing vertical ; shew that 
 the pressures upon them are respectively proportional to 
 l^m^ — mj,^, mj, — Z^m and ml^~ mj,, — and find each pressure. 
 
 46. If through the centre of gravity of each of the faces 
 of any polyhedron there act a force, in direction perpendicular 
 to the face and in magnitude proportional to its area, the 
 system will be in equilibrium, supposing all the forces to act 
 inwards or all to act outwards. 
 
 47. A frame formed of four uniform rods of the length (a) 
 connected by smooth hinges is hung over two smooth pegs 
 
 in the same horizontal line at a distance -?= , the two pegs 
 
 V2 
 
 being in contact with different rods. Shew that in the position 
 
 of equilibrium each angle = 90°. 
 
 Is the equilibrium stable or unstable ? 
 
 48. A heavy triangle ABO is suspended from a point 
 by three strings, mutually at right angles, attached to the 
 angular points of the triangle ; if ^ be the inclination of the 
 triangle to the horizon in its position of equilibrium, then 
 
 a 3 
 
 cos a = —rj- ^ TT . 
 
 V(l + sec A sec B secC; 
 
 49. From a right cone, the diameter of whose base is 
 equal to its altitude, is cut a right cylinder the diameter of 
 
MISCELLANEOUS EXAMPLES IN STATICS. 317 
 
 whose base is equal to its altitude, — ^their axes being in the 
 same line, and the base of the cylinder lying in the base of 
 the cone ; from the remaining cone a similar cylinder is cut, 
 and so on, indefinitely ; shew that the distance of the centre 
 of gravity of the remaining portion from the base of the cone 
 is ^ altitude of cone. 
 
 50. A uniform rod of length I is cut into three pieces 
 a, b, c, and these are formed into a triangle ; when the triangle 
 is placed in unstable equilibrium, resting with its plane ver- 
 tical and one of its angular points upon a smooth horizontal 
 plane, find the angle which the uppermost side makes with 
 the horizon ; — and shew that if a, )S, 7 be the three angles cor- 
 responding to the several cases of a, b, c being the uppermost 
 side, then 
 
 (Z+a)-tana+ (Z + &)tan y8+ (Z + c) tan7=0. 
 
 51. A string of equal spherical beads is placed upon a 
 smooth cone (2a) having its axis vertical, the beads being just 
 in contact with each other, so that there is no pressure be- 
 tween them. Find the tension t of the string ; and deduce 
 the limiting value T, when the number of beads is indefinitely 
 great. 
 
 Result. If Tr=sum of the weights of the beads 
 
 IF cot a Wcota 
 
 'lie ' 
 
 52. A weight is supported on a rough inclined plane (a) 
 by a force exactly equal to it. Shew that the direction of the 
 force may be changed through an angle 4tan"V without 
 disturbing the equilibrium of the weight, — provided that 
 
 tan '/t IS not < - nor > - - a. 
 
318 PEOBLEMS. 
 
 53. An even number of equal and uniform spherical 
 balls are slung in contact with each other on a fixed smooth 
 cylinder, whose axis is horizontal, by means of a string 
 which passes through smooth grooves pierced from the points 
 of contact of adjacent balls to the centres of the respective 
 balls. If the balls entirely surround the cylinder, and the 
 tension of the string be such that there is no pres'sure between 
 the fixed cylinder and the lowest ball which touches the 
 cylinder at its lowest point, shew that the pressure between 
 the cylinder and the highest ball is four times the weight of 
 each hall, 
 
 54. Three particles are connected by strings so as to 
 form a triangle and they are mutually repulsive ; shew that if 
 one particle be suddenly annihilated the tension of the string 
 connecting the other two will remain unaltered. 
 
 55. The particles of two circular discs repel each other 
 with a force varying as the distance. An endless elastic 
 string passes round their circumferences crossing between 
 them. If the discs were held in contact, the string would be 
 unstretched, and the resultant repulsion would be equal to the 
 modulus of elasticity. Shew that for equilibrium 
 
 sin 6 (sin 6 — 6cos6) = — , 
 
 where 26 is the i between the radii of either disc at the points 
 where the string leaves it. 
 
 56. Two uniform beams whose lengths are a and c are 
 capable of moving about hinges at their extremities placed 
 in the same horizontal plane. Another beam h is hinged to 
 their other extremities so that the system is above the hori- 
 zontal plane. If there be eqflilibrium the difference between 
 
MISCELLANEOUS EXAMPLES IN STATICS. 319 
 
 tlie lengths of the beam will be proportional to the difference 
 between the tangents of the angles which they make with 
 the horizon. 
 
 57. Two equal beams AB, AC, connected by a hinge 
 at A, are placed in a vertical plane with their extremities 
 B, C, resting on a horizontal plane ; they are kept from falling 
 by strings connecting B and C with the middle points of the 
 opposite sides ; shew that the ratio of the tension of either 
 string to the weight of either beam = | ^^ (8 cot" 6 + cosec" ff), 
 6 being the inclination of either beam to the horizon. 
 
 58. A uniform beam is supported upon the circumference 
 of a circle of radius »• In a vertical plane, by means of a string 
 of given length c, fastened at one end to the highest point of 
 the circumference, at the other end to one extremity of the 
 beam ; find the length of the beam that the string may be 
 horizontal. 
 
 JtewU. Length of beam 
 
 -m- 
 
 59. If the sector of a circle balance about the chord of 
 the arc, prove that, 2a being the angle of the sector, 
 
 2 tan a— 3a. 
 
 60. Two spheres of densities p, tr and radii a, b rest In 
 a paraboloid whose axis Is vertical, and touch each other at 
 the focus,— shew that pW = (T'b"'. Also If W, W be their 
 weights, and B, B' the pressures on the paraboloid at the 
 points of contact, 
 
 W W'~2\W' w)' 
 
 61. Two weights of different material are laid on an 
 inclined plane, connected by a string extended to its fuU 
 
320 ^ PROBLEMS. 
 
 length, inclined at an i.Q to the line of intersection of the 
 inclined plane with the horizon; if the lower weight be on 
 the point of motion, find the magnitude and direction of the 
 force of friction on the upper weight. 
 
 62. An endless string hangs at rest over two pegs in the 
 same horizontal plane, with a heavy puUy in each festoon of 
 the string J — if the weight of one puUy be double that of the 
 other, shew that the angle between the portions of the upper 
 festoon must be > 120°. 
 
 63. Two uniform beams loosely jointed at one extremity 
 are -placed upon the smooth arc of a parabola, whose axis is 
 vertical and vertex upwards. If I be the semi-latus-rectum 
 of the parabola, and a, h the lengths of the beams, shew that 
 they will rest in equilibrium at right angles to each other, 
 if Z (a + h) {a* + &*)* = aV, — and find the position of equili- 
 brium. 
 
 64. A heavy ring- hangs loose upon a fixed horizontal 
 cylinder, and is pulled by a string at its lowest point parallel 
 to the axis of the cylinder : find the limiting position of rest 
 when the coefficient of friction is given; — and shew that if 
 the coefficient of friction exceed a certain value, no force so 
 applied can make the ring slide. 
 
 65. A rod of length a is placed horizontally between 
 two pegs whose distances from opposite ends are respectively 
 ^a and Ja; if weights w and 3w be suspended from the ends 
 of the rod, find the tendency to break at any point of the 
 rod, and shew at what point it is the greatest. 
 
 66. A uniform rod (of length c) has smooth rings at 
 each extremity, one of which slides on a fixed vertical rod, 
 and through the other passes a fine string which is fixed at 
 
MISCELLANEOUS EXAMPLES IN STATICS. 321 
 
 two points A, B in tlie same horizontal line, the length of the 
 string being = 2 . AB; prove that the a which the rod makes 
 
 with the horizon in a position of equilibrium = cos~* ; 
 
 h being the distance of the fixed rod from the middle point 
 of AB and AB = 2a. 
 
 67. Five equal rigid heavy rods (each of weight W) 
 hinged together so as to form a regular pentagon ABODE, 
 are set in a vertical plane with one of them OB resting on a 
 horizontal table, and the form of the regular figure is pre- 
 served by help of an inextensible string connecting the hinges 
 B and E. Shew that the tension of the string 
 
 = iTr(tan54'' + 3tanl8''). 
 
 68. A string of length 1 is laid over two sniooth pegs 
 which are in the same horizontal line and at a distance a 
 from each other. Two unequal heavy particles, which attract 
 each other with forces varying as the distance, are attached, 
 one to each end of the string: shew that the inclination {0) 
 of either portion of the string to the horizon is given by the 
 equation 
 
 a tan — h = (l—a)sm 0, 
 
 where 25 = (the sum of the weights) -H (the attraction of the 
 particles at the unit of distance). 
 
 69. Four equal particles are mutually repulsive, the law 
 of force being that of the inverse distance. If they be joined 
 together by four inextensible strings of given length so as to 
 form a quadrilateral, — prove that when there is equilibrium, 
 the four particles lie in a circle. 
 
 70. A particle is at rest on a smooth vertical circle 
 \inder the action of gravity, and a force varying as the dis- 
 
 p. M. 21 
 
322 PROBLEMS. 
 
 tance from the extremity of a horizontal diameter, — ths 
 absolute force being such that the attraction on a particle 
 placed at the centre equals gravity: — shew that the particle 
 will rest half-way between the centre of force and the lowest 
 point of the circle, — and find the pressure on the curve. 
 
 71. A imiform bar is bent so as to form a triangle, and 
 the system rests on a smooth horizontal cylinder, whose 
 radius is nearly equal to that of the inscribed circle, — shew 
 that there will be no pressure on the greatest side a, and that 
 its inclination to the vertical will be 
 
 tan-' "^^"-2^> 
 
 (6~c) (2a-s)' 
 
 r being the radius of the cylinder, a, h, c the sides of the 
 triangle and 2s=a + b + c. 
 
 72. A heavy rod is placed in any manner resting on two 
 points of a rough horizontal curve, and a string attached to 
 the middle point G of the chord is pulled in any direction, so 
 that the rod is on the point of motion. Prove that the locus 
 of the intersection of the string with the directions of the 
 frictions at the points of support is an arc of a circle and 
 a part of a straight line. 
 
 Find also how the force must be applied that its inter- 
 sections with the frictions may trace out the remainder of the 
 circle. 
 Mouth and Watson's Senate-House Problems for 1860, p. 26. 
 
 73. ABGD is a quadrilateral, the intersection of the 
 diagonals; P, Q points in BD, AO such that QA= 00 and 
 PB= OB. Prove that the centre of gravity of the quadri- 
 lateral coincides with that of the triangle OPQ. 
 
 This simple and elegant construction for the centre of gravity of a plane 
 quadrilateral is given in the Quarterly Journal of Mathematics, Vol, vi. p. 127.- 
 
DYNAMICS. CHAP. I. 323 
 
 DYNAMICS. 
 
 INTRODUCTION. CHAPTER I. 
 
 1. A railway train travels over 150 miles in 5h, 40 m. 
 What is its average velocity in feet per second ? 
 
 Semlt, 38'S nearly. 
 
 2. What is the velocity of a particle which describes 
 4"38 miles in 31' 50":— a foot and a second being the respec-* 
 tive units of space and time ? 
 
 3. lif, F be the numerical values of any the same acce- 
 leration referred to units of time and space t,s; t, a respec- 
 
 tively, shew that F= - f-]f. 
 
 What would be the numerical value of the accelerating 
 force of gravity, if a mile and an hour were the units of space 
 and time? 
 
 4. If V, v' be two component velocities of a particle, and 
 a the angle between their directions, the resulting velocity is 
 = '\/{iy'+ v'^+ 2vv' cos a). 
 
 5. If the unit of pressure (or statical force) be 1 lb. and 
 the unit of accelerating force be the force which in a second 
 generates a velocity of one foot per second, what is the unit 
 of mass ? 
 
 Semlt. The mass of a weight of 32-21b8, 
 
 6. If the area of a field of ten acres be represented by 
 100, and the acceleration of a heavy falling particle by 58|, 
 lind the unit of time. 
 
 21—2 
 
324 PROBLEMS, 
 
 7. In the equation w = mg, what must be the relation 
 between the units of time and space, in order that the unit 
 of mass may be the mass of a unit of weight? 
 
 8. Shew frojn the second Jaw of motion that if a system 
 of particles subject to gravity be projected simultaneously 
 from a point in directions which all lie in one plane, the locus 
 of the particles at any subsequent instant will be a parallel 
 plane. 
 
 9. If the unit of weight be 1 oz., and one cubic foot of 
 the substance of standard density weigh 162 lbs., what must 
 be the unit of linear measure, that the formula W= Vpg may 
 be true, g being equal to 32 feet ? 
 
 SesvM. i inches. 
 
 10. In the equation of relation P= mf (Art. 42) , sup- 
 posing the unit of force to be 5 lbs. and the unit of accelera- 
 tion, referred to a foot and a second as units, to be 3, — ^find 
 the unit of mass. 
 
 Sesult. The unit of mass is the mass of 53f lbs. nearly. 
 
 11. The radius of the earth at the equator is 39G2"8 
 piles, and it makes a complete revolution about its axis in 
 23 A. 56 m.; find the velocity of a point at the equator in feet 
 per second. 
 
 Result. 1526 nearly. 
 
 12. If the accelerating efiect of gravity be numerically 
 represented by 9660, a yard being the linear unit, find the 
 unit of time. 
 
 Remit. Half a minute. 
 
 13. If a body weighing 30 lbs, be moved by a constant 
 force which generates in it in a second, a velocity of 50 feet 
 
DYNAMICS. CHAP. I. 32^ 
 
 per seconJ, find what weight the forde would statically sup- 
 port. 
 
 Se»uU. 46-77 lbs. nearly. 
 
 14. The wind blowing exactly along a line of railway, 
 two equally quick trains, moving in opposite directions, have 
 the steam track of the one twice as long as that of the other ; 
 compare the velocities of the trains and of the wind. 
 
 Eemli. Velocity of the train =3 times that of the wind. 
 
 15. If X, f^ be the measure of the accelerating effect of a 
 force when m + n and m — n seconds are the respective units of 
 time, and a and b feet the respective units of distance, — shew 
 
 that the measure becomes - (V/^o+V^J)'',— provided 2m seconds 
 
 be the units of time, and c feet the unit of distance. 
 
 16. A point, moving with a uniform acceleration, de- 
 scribes 20 feet in the half-second which elapses after the first 
 second of its motion ; compare its acceleration f with that of 
 a falling heavy particle: — and give its numerical measure, 
 taking a minute as the unit of time, and a mile as that of 
 space. 
 
 SesuU. (i)/:ar=l : 1 nearly, (ii) /=21^. 
 
 17. A pressure P produces an accelerating effect / on a 
 mass m, determine the relation between P, m and /; the unit 
 of pressure being 1 lb. the unit of mass the mass of a cubic 
 foot of water, and the Unit of acceleration the acceleration 
 produced by gravity. 
 
 Semlt P=62-5.m.f. 
 
 18. If a point be situated at the intersection of the per-. 
 pendiculars drawn from the angular points of a triangle to 
 the sides respectively opposite to them, and have component 
 
326 PROBLEMS. 
 
 velocities represented in magnitude and direction hj its dis- 
 tances from the angular points of the triangle, — prove that 
 its resultant velocity will tend to the centre of the circle 
 circumscribing the triangle, and will he represented hy twice 
 the distance of the point from the centre. 
 
 19. If a he the distance at any time between two points 
 moving uniformly in one plane, V their relative velocity, and 
 u, V the resolved parts of V in and perpendicular to the direc- 
 tion of a, shew that their distance when they are nearest to 
 
 each other is -^ , and that the time of arriving at this near- 
 est distance is = t^„ . 
 
 20. A straight rod moves in any manner in a plane; 
 prove that at any instant the directions of motion of all its 
 particles are tangents to a parabola. 
 
 21. A person travelling eastward at the rate of 4 miles an 
 hour, observes that the wind seems to blow directly from the 
 north ; on doubling his speed the wind appears to come from 
 the north-east ; determine the direction of the wind, and its 
 velocity. 
 
 Semlt. The trae direction of the wind is from the north-west — and its 
 velocity is 4*^2 miles an hour. 
 
 22. The measures of an acceleration and a velocity when 
 referred to (a + b) ft., (m + n)" and (a — h) ft., (m — n)" respec- 
 tively, are in the inverse ratio of their measures when referred 
 to (a — 6) ft., (m — n)" and (a + J)ft., (m + n)"; their measures 
 when referred to a ft., m" and b ft., n'' are as ma : nb, shew 
 that 
 
COLLISION. CHAP. II. 327 
 
 COLLISION. CHAPTER 11. 
 
 1. What must be the elasticity of two balls A, B in 
 order that A impinging directly upon B at rest may itself be 
 reduced to rest by the impact ? 
 
 « , A 
 Mexim. e=-^. 
 M 
 
 2. A man can pull a boat with three times the velocity 
 of the stream — at what angle to the stream must the boat 
 be rowed in order that he may land at a point directly oppo- 
 site his starting place ? 
 
 Itesult. At an angle with the stream— cos~^ ^. 
 
 3. A ship sails N.W. at the rate of 9 knots per hour, 
 and is drifted S.S.W. by the current at the rate of 2 knots 
 an hour — find the actual speed and direction of motion. 
 
 Semlt. 'B.er Bpeed = v85-18,y/2-iy2 knots an hour, — her direction 
 
 ,9-2cog— . 
 makes an angle cot"' I ] to the west of north-west. 
 
 \ 2 sin -J / 
 
 !sm-g 
 
 4. A ball of 9 ounces moving with a velocity of 7 feet 
 a second impinges directly upon a ball of 12 ounces moving 
 with a velocity of 5 feet a second in the opposite direction ; 
 find the change in the velocity and momentum of each ball, 
 supposing them inelastic. 
 
 5. ' Under what conditions will the .velocities of two balls 
 A, B impinging directly upon each other, be interchanged 
 after impact? 
 
 Setult. If the balls be eqnal and Ike elastidty perfect. 
 
328 . PEOBLEMS. > 
 
 6. Two balls A, B are moving in directions at right 
 angles to each other with the same velocity, the line joining 
 their centres at the instant of impact being in direction of ^'s 
 motion; find the velocity and direction of motion of each 
 after impact (elasticity = 6). , 
 
 Remit. In the formulse of Art. 58 write o=0, /3=90'', «=v. 
 
 7. Two bodies of masses 2,4 and ^A are moving with 
 the same velocity in directions making angles 45° and 30" 
 with the common tangent at the point of impact. Find the 
 direction and velocity of the centre of gravity. 
 
 8. A, B are two equal and perfectly elastic spheres; 
 A moving with a given velocity impinges on B at rest, the 
 direction of A^s motion before impact making an angle of 60° 
 with the straight line which joins their centres at the instant 
 of impact ; determine the directions and velocities of A and B 
 after impact. 
 
 9. Compare the velocity of a place at the earth's equa- 
 tor arising from the earth's rotation, with the velocity of the 
 earth in her orbit about the sun ; assuming the earth's radius 
 = 4000 miles, the radius of the earth's orbit = 95000000 miles,--' 
 and the length of the year = 365^ days. 
 
 JtesuU. 1 : 65 nearly. 
 
 10. A ball A impinges directly with a given velocity 
 upon another ball B at rest ; if the vis viva before impact 
 be n times the vis viva after impact, find their common 
 elasticity. 
 
 A+B-nA 
 
 Jies-ulU «"=- 
 
 nB 
 
 11. A ball A moving with a given velocity impingea 
 directly upon a ball B at rest; and B afterwards impinges 
 
COLLISION. CHAP. II. 329 
 
 Upon at rest J determine the velocity communicated to C. 
 li A and Cbe of given mass and 5 variable,. shew that C"s 
 velocity will be greatest when B^ = A.G. 
 
 Apply the formulas of Art. 58. 
 
 12. A ball A strikes a ball B at rest, the direction of 
 ^'s motion before impact being 45° inclined to the line AB; 
 find the velocity- and direction of motion of each after impact, 
 and the condition that they may move at right angles to each, 
 other. 
 
 13. A perfectly elastic ball acted on by no force, is pro- 
 jected from the focus of an ellipse and impinges upon the. 
 curve; it will return to the. focus again, in the same time, 
 whatever be the direction of projection. 
 
 14. Two planes make an angle of 5° with each other, 
 and a perfectly elastic body is projected against one of them 
 at an angle of 105°; how many reflexions will take place 
 towards the angle where the planes meet ? 
 
 Sesult. Three. 
 
 15. A ball A impinges obliquely on another ball B at 
 rest, and after impact the directions of motion of A and B 
 make equal angles (a) with A^s previous motion : find ,a, and 
 shew that if the masses of the balls be equal and e the 
 mutual elasticity, a = i&vT^Je. 
 
 16. A smooth table has a smooth rim in the form of a 
 regular hexagon ; shew that an inelastic ball, projected along 
 one side of the hexagon, performs n complete revolutiona in. 
 (2°"— 1) time of describing the first side. 
 
 17. Two imperfectly elastic balls, equal in size, but Un- 
 equal in mass, are placed between two perfectly" hard parallel 
 
330 PEOBLEMS. 
 
 planes, to wMch the line joining the centres of the balls is 
 perpendicular, — each ball being initially at a distance from 
 the plane nearest to it, inversely proportional to its mass. 
 The balls approach each other with velocities inversely pro- 
 portional to their masses ; prove that every impact will take 
 place at the same point as the first does. 
 
 18. Two balls, of elasticity e, moving in parallel direc- 
 tions with equal momenta, impinge ; prove that if their direc- 
 tions of motion be opposite, they will move after impact in 
 parallel directions with equal momenta ; and that these direc- 
 tions will be perpendicular to the original direction if their 
 common normal is inclined at an angle t&vT^Je to that 
 direction. 
 
 19. A ball of elasticity e is projected along a horizontal 
 plane in an equilateral triangle, and after reflexion at two 
 sides it impinges perpendicularly on the third. Shew that the 
 
 , p. ., ^ _,\/3e(l-e) 
 angle ot mcidence was tan — . 
 
 1 "T~ o6 
 
 20. If M, V be the velocities before direct impact of two 
 balls A, B, — u', v' their velocities after impact, shew that 
 
 Au'+Bv'= Au"+ Bv"-[- —^ (1 - e') (« - v)\ 
 
 21. A body whose elasticity is e is projected from a point 
 m the circumference of a circle, and after three rebounds from 
 the circumference returns to the point from which it was pro- 
 jected ; shew that the direction of projection is inclined to 
 the radius of the circle at an angle = tan"'(e'). 
 
COLLISION. CHAP. II. 831 
 
 22. A ball projected from a point in one side of a billiard 
 table returns to tlie point of projection after striking each side 
 in succession ; find the direction of projection, and shew that 
 if it ever returns to its original position it does so after the 
 first circuit. 
 
 23. Two equal balls (A, A), moving with equal velocities 
 in directions passing through the centre of a third ball C, 
 impinge upon it and upon one another simultaneously ; find 
 the ratio of the masses of the balls, that after impact the direc- 
 tions of motion of the two balls may be perpendicular to that 
 of the third, the coefficients of elasticity being ^. 
 
 BesuU. C=iA. 
 
 24. A ball A Impinges upon a ball B at rest ; find the 
 direction of the line joining the centres of A and B, in order 
 that they may after impact move in directions making equal 
 angles with the original direction of ^'s motion. 
 
 ReivM. "With the notation of Art. (58) we must have 
 ^ , B-A + ieB 
 
 *^°«=-Xkb — 
 
 25. If ABO be a triangle and D, E, F the points where 
 the circle inscribed in it meets the sides BO, GA, AB respec- 
 tively; shew that if a ball, of elasticity e, be projected from 
 Z> so as to strike AOva.E and then rebound to Fy 
 
 AE=^e.OE. , 
 
 If the ball return to -D, AB=e.AO. 
 
 26. Two equal balls (of elasticity e) start at the same 
 instant with equal velocities from the opposite angles of a 
 square along the sides and Impinge ; determine the angle 
 between their directions after the impact. 
 
 JUf^alt. tan,-' = — -j . 
 
 * ~6 
 
332 PKOBLEMS. 
 
 S7. Three equal smooth balls rest on a horizontal table 
 and each is in contact with the other two ; if one of them 
 receive a blow at a given point in the plane passing through 
 the centres of the balls, determine the direction of its motion 
 after impact. 
 
 28. Two particles connected by an inextensible string 
 are projected in given directions in one plane with given velo- 
 cities ; determine their motions immediately after the string 
 becomes tight, 
 
 29. A body of elasticity e is projected along a horizontal 
 plane from the middle point of one of the sides of an isosceles 
 right-angled triangle, so as after reflexion at the hypotenuse 
 and remaining side to return to the same point; shew that 
 the cotangents of the angles of reflexion are e-\-l and e + 2 
 respectively. 
 
 30. The tangents pf the angles of a triangle ABG are 
 in geometrical progression, tan 5 being the mean proportional; 
 and a ball is projected in a direction parallel to the side GB, 
 so as to strike the sides AB, BG successively. Shew that if 
 its course after the first impact be parallel to A G, its course 
 after the second wQl be parallel to BA :^-and that if e be the 
 modulus of elasticity, 
 
 e*M-e"* = sec^. 
 
 31. A ball is projected from the middle point of one side' 
 pf a billiard table so as to strike in succession one of the sides 
 adjacent to it, the side opposite to it, and a ball placed in the 
 centre of the table : shew that if a, h be the lengths of the 
 sides of the table, e the elasticity pf the ba.ll, the inclination 
 
COLLISION. CHAP. II. 333 
 
 of the direction of projection to the side a of the table from 
 which it is projected must be 
 
 a\l+ej 
 
 32. A smooth inelastic ball, — mass m, — is lying on a 
 horizontal table in contact with a vertical wall, and is struck 
 by another ball, — ^mass m' — moving in a direction perpen- 
 dicular to the wall, making an angle (a) with the common 
 normal at the point of impact ; shew that if 6 be the angle 
 through which the direction of motion of the striking ball is 
 turned, 
 
 cot6 cot a = — 1-1. 
 m 
 
 33. An elastic ball is projected from a point in one of 
 the sides of a square billiard table so as to describe an in- 
 scribed square ; prove that if e be the mutual elasticity of the 
 cushions and ball, the time of describing the square is 
 
 l-e' 1 
 1 — e'i e^ 
 time of describing the first side. 
 
 34. A particle, of elasticity e, is projected from the middle 
 point of one side of a square, in a direction making an z ^ 
 with it ; — shew that if the ball strike the four sides in order, 
 6 must lie between 
 
 ^ _i 2e (1 -t- e) , , . -1 2 (1+ e) 
 tan^ , , \, , ' and tan — ;— — - . 
 1+. (l+e)" 2 + 6 
 
 , 35. Two billiard balls are lying in contact on the table ; 
 ^n what direction must one of them be struck by a third, so 
 as to go off in a given direction ? 
 
 X 
 
334 PEOBLEMS. 
 
 36. A row of elastic balls A, B, C, ... P, are at rest; if 
 A be made to impinge directly with given velocity upon B, 
 then B on C with the velocity acquired, on D, and so on, 
 find the velocity of P. 
 
 Shew that if A and P be of given magnitude, but B, C... 
 capable of being changed, the velocity communicated to P 
 will be greatest when the masses of the balls are in geo- 
 metrical progression. 
 
 And if the number of balls interposed between A and P 
 become indefinitely great, then the velocity acquired hy 
 
 P= 1 / ( p j . original velocity of A. 
 
 ACCELERATED MOTION. CHAPTER III. 
 
 1. A BODY is projected upwards with a velocity u, and 
 after rising through a space s, has a velocity v ; shew that 
 
 I)' = m" — 2ys. 
 
 If the velocity of projection is 8^, find the time in which 
 the body rises through the height Ug. 
 
 2. A particle of elasticity ^ drops through 16 feet, and 
 then rises after impact on a horizontal plane. Find the velo- 
 city after rising 3 feet, and the time of this ascent : force of 
 gravity being taken to be 32 feet. 
 
 Remit. Velocity =8 feet, and tte time=^ second, 
 
 3. A particle moves over 7 feet in the first second of the 
 time during which it is observed, and over 11 and 17 feet in 
 the 3rd and 6th seconds respectively. Is this consistent with 
 
ACCELEEATION. CHAP. III. 335 
 
 the supposition of its being subject to the action of a uniform 
 force ? 
 
 Mesult. Yea. 
 
 4. A weight Q is drawn along a smooth horizontal table 
 by a weight P hanging vertically, find (1) the acceleration 
 of P, (2) the acceleration of the centre of gravity of P 
 and Q. 
 
 P 
 
 Besult. (i) Acceleration of P = -p — q9- (ii) Of the centre of gravity 
 
 J. 
 
 f p \i PQ 
 
 [ „ q ) g vertically, and = ,p,n\i 9 'horizontally. 
 
 5. A constant force (/) acts upon a body from rest during 
 3 seconds, and then ceases. In the next 3 seconds it is found 
 that the body describes 180 feet. Find both the velocity {v) 
 of the body at the end of the 2nd second of its motion and the 
 numerical values of the accelerating force (1) when a second, 
 (2) when a minute is taken as the unit of time. 
 
 MmiU. J)=iO, (i) /=20. (ii) /= 72000. 
 
 6. A force which can statically support 50 lbs. acts uni- 
 formly for one minute on a body, the weight of which is 
 200 lbs. ; find the velocity and momentum acquired by the 
 body. 
 
 7. A body acted on by a uniform force is found to be 
 moving at the end of the first minute from rest with a velocity 
 which would carry it through 10 miles in the next hour. 
 Compare this force /with that of gravity g. 
 
 JtesiiU. f:g=l:Ul nearly. 
 
 8. If the force of gravity be taken as the unit of force, 
 and a rate of ten miles an hour as the unit of velocity, what 
 must be the units of time and space? 
 
 /11\" 121 
 
 Besiat. Unit of time= f gj j , vinit of space= -^ feet. 
 
336 . PROBLEMS. 
 
 9. A bullet fired directly into a block of wood will pene- 
 trate a inches : find what proportion of its velocity it would 
 lose in passing through a board of the same wood one inch 
 thick, supposing the resistance uniform. 
 
 10. A particle slides down a rough inclined plane (a) ; 
 find the acceleration /. 
 
 f=g (sin a- II, cos o). 
 
 11. If a weight of ten pounds be placed upon a plane 
 which is made to descend with a uniform acceleration of 10 
 feet per second, what is the pressure upon the plane? 
 
 Hemlt. 6-875 lbs. 
 
 12. A body falling in vacuo under the action of gravity 
 js observed to fall through 144"9 feet and 177*1 feet in two 
 successive seconds ; determine the accelerating force of gravity, 
 and the time from the beginning of the motion. 
 
 Besidt. g= 32"2, and the first of the two seconds spoken of is the fifth from 
 the beginning of motion. 
 
 13. The velocity generated by a gun in a bullet of 1 oz. 
 is 1000 feet per second ; supposing that the bullet described 
 the length of the barrel in Jg- of a second, and that the force is 
 uniform, find the acceleration and moving force {f, F). 
 
 Result. /= 10000 feet per second, 
 J''=19'41bs. nearly. 
 
 14 A body falling vertically is observed to describe 
 112"7 feet in a certain second; how long previously to this 
 has it been falling ? 
 
 Hesult. Three seconds, 
 
ACCELERATION. CHAP. III. 337 
 
 15. A person drops a stone into a well, and after f hears 
 it strike the water ; find the depth (a;) to the surface of the 
 water (assuming velocity of sound = Z^.g nearly). 
 
 Jteiult. Find x fropa the equation 
 
 x + Z5,,j2gx=Z5gt. 
 
 16. A balloon ascends with a uniformly accelerated 
 velocity so that a weight of 1 lb. produces on the hand of the 
 aeronaut sustaining it a downward pressure equal to that 
 which 17 oz. would produce at the earth's surface; find the 
 height which the balloon will have attained in one minute 
 from the time of starting, not taking into account the varia- 
 tion of the accelerating effect of the earth's attraction. 
 
 Result. 1 207 -5 yards, taking fl' = 32 -2. 
 
 17. AB is the vertical diameter of a circle ; through A 
 the highest point any chord AG is drawn, and through G a 
 tangent meeting the tangent at B in the point T. Shew that 
 
 the time of a body's sliding down GTcn -^n • 
 
 18. A particle uniformly accelerated describes 108 and 
 140 feet in the 5th and 7th seconds of its motion : — find the 
 velocity of projection and the numerical measure of the ac- 
 celeration. 
 
 19. Shew how to place a plane of given length in order 
 that a body may acquire a given velocity by falling down it. 
 
 20. Prove that the locus of the points, from which the 
 times down equally rough inclined planes to a fixed pomt 
 vary as the lengths of the planes,' is a right circular cone. 
 
 21. In a parabola whose axis is vertical, a tangent is 
 
 drawn at any point P cutting the axis produced in T; shew 
 
 22 
 p. M. ^ ^ 
 
338 PROBLEMS. 
 
 that if gravity alone acts, the time of descent down TP bears 
 a constant ratio to the time of descent from T to the focus. 
 
 22. APB, AQC are two circles with their centres in the 
 same vertical line ABC, and touching each other at their 
 highest points. If APQ, Apg^ be any two chords, the times 
 of descent down PQ, pq^ from rest at P and p are ecLual. 
 
 23. A particle is moving under the action of a uniform 
 force, the accelerating effect of which is f: if u be the arith- 
 metic mean of the first and last velocities in passing over any 
 portion h of the path, and v the velocity gained, shew that 
 
 uv =fh. 
 
 24. In what time will a force which would support a 
 5 lb. weight move a mass of 10 lbs. weight through 50 feet 
 along a smooth horizontal plane, and what will be the velo- 
 city acquired? 
 
 25. If a body subject to a uniform acceleration describes 
 36 feet, whilst its velocity increases from 8 to 10 feet per 
 second, how much farther will it be carried before it attains 
 a velocity of 12 feet per second? 
 
 26. A heavy body is projected up an inclined plane, 
 inclined at 60° to the horizon, with the velocity which it 
 would have acquired in falling freely through a space of 
 12 feet, and just reaches the top of the plane; find the alti- 
 tude of the plane, the coefficient of dynamical friction being 
 _ J|_ 
 
 ~V3" 
 
 SeguU. 9 feet, 
 
 27. Two bodies uniformly accelerated, in passing over 
 the same space, have their respective velocities increased 
 
ACCELERATIOxX. CHAP. III. 339 
 
 from 5 to 7 and from 8 to 10,— compare the accelerating 
 forces, and the respective times of describing the space. 
 * -v » % -s -s 
 
 28. AP, A Q are two inclined planes of which AP is 
 rough {fi.= t&nPAQ) and AQ h smooth, AP lying above 
 A Q : shew that if bodies descend from rest at P and Q, they 
 will arrive at A, (i) in the same time if PQ be perpendicular 
 to A Q, (ii) with the same velocity if PQ be perpendicular to 
 AP. 
 
 29. An engine whose power is sufficient to generate a 
 velocity of 150 feet a second in a mass m (which is its own 
 
 mass) is attached to a carriage, mass = — , by means of an 
 
 inelastic weightless chain 3 feet long; this carriage again in 
 
 exactly the same way to another mass = -^ ; this to a third 
 
 mass = -3 . The engine and' carriages are in contact when 
 
 the train starts; shew that the last carriage will hegin to move 
 with a velocity = 33 feet per second nearly. 
 
 30. A body P descending vertically draws another body 
 Q up the inclined plane formed by the upper surface of a 
 right-angled wedge which rests on a smooth horizontal table; 
 find the force F necessary to prevent the wedge from sliding 
 along the table. 
 
 \sma — P 
 
 Besult. F= Qg cos o 
 
 P+Q, 
 
 31. A uniform string hangs at rest over a smooth peg. 
 Half the string on one side of the peg is cut off: shew that 
 the pressure on the peg is instantaneously reduced to two- 
 thirds its previous amount. 
 
 22—2 
 
340 PROBLEMS. 
 
 32. A smooth wedge (of io) on a horizontal plane is 
 moved from rest with a uniform acceleration; find the direc- 
 tion and amount of the acceleration that a heavy particle 
 placed on its inclined plane surface may be in equilibrium 
 relative to it. 
 
 Besidt. The wedge must move in a principal plane with an acceleration 
 =g tan a. 
 
 33. Find the locus of points from which inelastic parti- 
 cles may be let fall on a smooth inclined plane, so as always 
 to have the same velocity on arriving at the same horizontal 
 line in the plane. 
 
 Remit. A plane passing through the giren horizontal line. 
 
 34. If a body is projected with velocity u in the direc- 
 tion of a uniform force f, and if v be the velocity and s the 
 space described at the end of time t, prove that 
 
 V — u 2s 
 
 f V + ' 
 
 = t. 
 
 The velocity of a body increases from 10 to 16 feet per 
 second in passing over 13 feet under the action of a constant 
 force; find the numerical value of the force. 
 
 35. Find- by geometrical construction or otherwise the 
 line of quickest descent, 
 
 (i) From a given straight line to a given point. 
 
 (ii) From a given point within a given circle to the 
 circle. 
 
 (iii) From a given circle to a given point within it. 
 
 (iv) From a given circle to a given straight line or to 
 another circle without it. 
 
ACCELERATION. CHAP. III. 341 
 
 (v) From a given circle to another giren circle either 
 within it or without it. 
 
 36. Two circles lie in the same plane, the lowest point 
 of one being in contact with the highest point of the other; 
 shew that the time of descent from any point of the former 
 to a point in the latter, down the chord passing through the 
 point of contact, is constant. 
 
 37. Two equal bodies connected by a string are placed 
 upon two planes which are inclined at angles a, /8, to the 
 horizon, and have a common altitude. Prove that the acce- 
 leration of their centre of gravity is 
 
 ^-.sm^-.cos'— g— . 
 
 38. A number of heavy particles start at once from the 
 vertex of an oblique circular cone, whose base is horizontal, 
 and fall in all directions down generating lines of the surface; 
 prove that they will at any subsequent moment lie in a sub- 
 contrary section. 
 
 39. Two bodies A and B descend from the same ex- 
 tremity of the vertical diameter of a circle, one down the 
 diameter, the other down the chord of 30°. Find the ratio of 
 ^ to -B when their centre of gravity moves along the chord 
 of 120°. 
 
 Result. A : B^Js + l : 1. 
 
 40. A series of particles slide down the smooth faces of 
 a pyramid, starting simultaneously from rest at the vertex; 
 shew that after any time t they are in a certain spherical 
 surface whose radius = \gf. 
 
 41. P pulls Q over a smooth puUy; — and Q in ascend- 
 ing as it passes a certain point A, catches and carries with it 
 
342 PROBLEMS. 
 
 a certain additional weight which makes it altogether heavier 
 than P; and on its descent the additional weight is again 
 deposited at A. Supposing no impulse to take place when 
 the weight is so caught up, and that Q in this manner oscil- 
 lates through an equal space on either side of A, — find the 
 additional weight, 
 
 42. If the weight attached to the free end of the string 
 in a system of puUies, in which the same string passes round 
 each pully, be m times that which is necessary to maintain 
 equilibrium, shew that the acceleration of the ascending 
 
 weight is ; . q, where n is the number of strings at the 
 
 ° mn +1 ° 
 
 lower block, and the grooves of the pullies are supposed 
 
 smooth. What is the tension of the string? 
 
 43. A weight W is connected with a weight P by a sys- 
 tem of n movable pullies, in which the string passing round 
 any pully has one end fixed and the other attached to the 
 pully next above it — the string to which Pis attached passing 
 round a fixed pully, and the strings between the pullies 
 being all parallel : — shew that the acceleration of W upwards 
 . _ TP- W 
 
 44. If 8 be the focus of a parabola whose axis is hori- 
 zontal and plane vertical, 8P the line of quickest descent 
 from 8 to the curve, shew that 8P is inclined at 60° to the 
 axis. 
 
 45. Two weights P, Q move on two planes inclined at 
 angles a, /3 to the horizon respectively, being connected by a 
 fine string passing over the common vertex, in a vertical 
 •plane which is at right angles to this common vertex, their 
 
ACCELEKATION. CHAP. III. 343 
 
 centre of gravity describes a straight line with uniform ac- 
 celeration equal to 
 
 Qsin/S — Psin a. 
 
 ^ Dill Aj — JL Bill «, /-=ri „ „„ ; ;^T 773 
 
 3 ,p^ QY ^^ + 2^<3 COS (a + /3) + Q'. 
 
 46. A heavy particle is projected directly up an inclined 
 plane (a) with velocity u, and is attached to the point of 
 projection by an inextensible string whose length is half the 
 distance a free particle would ascend : determine the time 
 which elapses before the particle returns to the point of pro- 
 jection. 
 
 47. Supposing the weights in Atwood's machine to bs 
 7 and 9 pounds and to rest on scale-pans without weight, find 
 the pressure on each scale-pan. 
 
 48. A body starts from rest under a uniform acceleration, 
 but at the commen(Jement of each successive second the ac- 
 celeration is decreased in a geometrical proportion (*• = ^) : — 
 
 shew that the space described in n seconds = i2n — S + —„j is, 
 
 — ^where s is the space described in the first second. 
 
 49. Two bodies whose weights are P and Q hang from 
 the extremities of a cord passing over a smooth peg ; if at the 
 end of each second from the beginning of motion F be sud- 
 denly diminished and Q suddenly increased by - th of their 
 
 original difference ; shew that their velocity will be zero at 
 the end of « + 1 seconds. 
 
 50. A string charged with n + m + l equal weights, fixed 
 at equal intervals along it, and which would rest on a smooth 
 inclined plane with m of the weights hanging over the top, 
 
344 PEOBLEMS. 
 
 is placed on the plane with the {m + 1)* weight just over 
 the top ; — shew that if a be the distance between each two 
 adjacent weights, the velocity which the string will have 
 acquired at the instant the last weight slips off the plane, 
 will be = ^fnag. 
 
 51. A fine inelastic thread is loaded with n equal par- 
 ticles at equal distances c from one another; the thread is 
 stretched and placed on a smooth horizontal table, perpen- 
 dicular to its edge, over which one particle just hangs ; find 
 the velocity of the system when the r"" particle is leaving the 
 table. 
 
 Hence shew that if a heavy string of length a be simi- 
 larly placed on a horizontal table, its velocity in falling off 
 will be = i/{ag). 
 
 Result. v,^=gc— . 
 
 n 
 
 52. A number n of equal balls connected by a string are 
 laid upon a smooth table, the string being stretched at right 
 angles to the edge of the table ; if one ball hanging over the 
 edge draws the others after it, determine the lengths of suc- 
 cessive portions of the string, that each may fall over at the 
 end of successive equal intervals of time. 
 
 Result. If a, be the length of string between the r* and (r + 1)*'' balls, 
 we must have ar=r^.a\, and if Vr be the velocity of the system when the »■*'' 
 
 ball is passing over the edge, », = r (r - 1) 
 
 V -In 
 
PROJECTILES. CHAP. IT. 345 
 
 PEOJECTILES. CHAPTER IV. 
 
 1. A particle, acted on by two equal centres of force 
 whicli vary as distance,— one repulsive and the other attrac- 
 tive, — will, however projected, describe a parabola. 
 
 2. A body is projected with a vertical velocity (16-7) 
 and a horizontal velocity (-8) ; prove that its distance from 
 the point of projection at the end of one second is one foot 
 {g = 32-2 feet). 
 
 3. If a body fall down an inclined plane (a), and another 
 be projected from the starting point horizontally along the 
 plane with velocity v, find the distance D between the two 
 bodies (i) after a given time t, (ii) after the first body has 
 descended through a given space s. 
 
 RemU. (i) D=vt. (ii) I)=v \/ 
 
 2s 
 gsiua' 
 
 4. Find the angle which the direction of a projectile 
 makes with the horizon at any point of its path, and deter- 
 mine its distance from a line drawn through the point of pro- 
 jection parallel to this direction. 
 
 Result. With the notation of Art. 88, Cor. 3, 
 
 gt 
 
 tan ^ tan a — , 
 
 V cos a 
 
 and z = distance required = v sin (o - 0) . < — ^j" cos ^ . <'. 
 
 5. If 6, ^ be the angles which the tangents to the curve 
 at the points P, Q of the path of a projectile make with the 
 horizon, the time of describing the arc P^cc tan 6 — tan <f>. 
 
 6. A body slides down an inclined plane of given hdght, 
 and then impinges upon an elastic horizontal plane ; what 
 
346 PROBLEMS. 
 
 must te the elevation of the inclined plane in order that the 
 range on the horizontal plane may be the greatest possible ? 
 
 Result. 45". 
 
 7. Having given the velocities at two points of the path 
 of a projectile, find the difference of their altitudes above a 
 horizontal plane. 
 
 8. If a ship is moving horizontally with a velocity Zg, 
 and a body is let fall from the top of the mast, find its velocity 
 and direction after 4". 
 
 Result. Velocity = f 5', inclination to the horizon =tan"'^ -. 
 
 o 
 
 9. A body is projected from the top of a tower with a 
 given velocity in a given direction ; find where it will strike 
 the ground. 
 
 10. A heavy particle is projected from one point so as 
 to pass through another not in the same horizontal line with 
 it; prove that the locus of the focus of its path will be a 
 hyperbola. 
 
 11. Particles are projected from the same point in a ver- 
 tical plane with velocities which vary aa (sin 6)~^, 6 being the 
 angle of projection ; the locus of the vertices of the parabolas 
 described is an ellipse — -whose, horizontal axis is double the 
 vertical axis. 
 
 12. Two heavy bodies are projected from the same point, 
 at the same instant, in the same direction, with different 
 velocities ; find the direction of the line joining them at any 
 subsequent time. 
 
 Result. It is always parallel to the direction of projection. 
 
PKOJECTILES. CHAP. IV. 347 
 
 13. An imperfectly elastic tall is projected from a point 
 between two vertical planes, the plane of motion being per- 
 pendicular to both ; shew that the arcs described between the 
 rebounds are portions of parabolas whose latera recta are in 
 geometric progression. 
 
 14. A body is projected vertically upwards from a point 
 A with a given velocity (m) ; find the direction (a) in which 
 another body must be projected with a given velocity {v) 
 from a point B in the same horizontal line with A, so as to 
 strike the first body. 
 
 Result. aina=-. 
 
 15. A ball is projected from a point in a horizontal plane 
 and makes one rebound; shew that if the second range is equal 
 to the greatest height which the ball attains, tan a = 4e : 
 
 a being the angle of projection and e the elasticity. 
 
 16. Particles are projected from the same point in the 
 same direction, but with different velocities ; find the locus of 
 the foci of their paths. 
 
 Remit. The straight line 3/+ci!oot2o = (Art. 88). 
 
 17. The greatest range of a rifle-ball on level ground is 
 1176"3 feet. Find the initial velocity of the ball, and shew 
 that the greatest range up an incline of 30° will be 784*2 feet 
 — neglecting the resistance of the air. 
 
 18. If a body be projected at an angle a to the horizon 
 with the velocity due to gravity in 1", its direction is inclined 
 
 at an angle - to the horizon at the time tan-, and at an 
 angle '""~ ■ at the time cot- . 
 
348 PEOBLEMS. 
 
 19. A body is projected from a given point A with a 
 given velocity and in a given direction. After a lapse of 
 m seconds another equal body is projected from the same 
 point so that the line joining the two bodies always passes 
 through A : shew that the paths of the two bodies and that 
 of their centre of gravity will be equal parabolas. 
 
 20. A perfectly elastic particle is projected with a given 
 velocity from a given point in one of two planes equally in- 
 clined to the horizon and whose line of intersection is hori- 
 zontal : determine the angle of projection in order that the 
 particle may after reflexion return to the point of projection, 
 and be again reflected in the same path. 
 
 Shew that each plane must be inclined at an angle - to 
 
 the horizon. 
 
 21. A particle projected with velocity v impinges per- 
 pendicularly on an inclined plane drawn through the point 
 of projection at an inclination a, — shew that the range on the 
 
 , 2u'' sin a 
 
 plane = — - — - — ^-r— . 
 '■ g l + 3sm''a 
 
 :{: H: ^ ^ ^ ^ * 
 
 22. A body is projected from a given point in a horizontal 
 direction with a given velocity, and moves upon an inclined 
 plane passing through the point. If the inclination of the 
 plane vary, the locus of the directrix of the parabola which 
 the body describes is a horizontal plane. 
 
 23. A body is projected horizontally with a velocity Ag 
 from a point whose height above the ground is 16^ ; find the 
 direction of motion (1) when it has fallen half-way to the 
 ground, (2) when half the whole time of falling has elapsed. 
 
 BemU. (i) ^=45". (ii) ^=tan-^-p. 
 
PROJECTILES. CHAP. IV. 349 
 
 24. A cylinder is made to revolye uniformly about its 
 axis, which is vertical, while a body descends under the 
 action of gravity, carrying a pencil which traces a curve On 
 the surface of the cylinder : if the surface of the cylinder be 
 unwrapped, what will be the nature of the curve? 
 
 Result. A parabola with axis vertical. 
 
 25. If a ball ,of elasticity | is let fall through a height h 
 on a plane whose inclination is 30°, shew that it will strike 
 
 the plane again at a distance — from the first point where it 
 strikes the plane. 
 
 26. If the initial velocity of a projectile be given, the 
 horizontal range is the same, whether the angle of projection 
 
 be — + a, or - — a. Prove this, and compare the times of 
 flight. 
 
 27. The velocities at the extremities of any chord of the 
 parabola described by. a body projected obliquely and acted 
 on by gravity, when resolved in a direction perpendicular to 
 the chord, are equal. 
 
 28. From the top of a tower two bodies are projected 
 with the same given velocity at different given angles of 
 elevation, and they strike the horizon at the same place; find 
 the height of the tower. 
 
 29. Having given the velocity and direction of projec- 
 tion of a projectile, determine by a geometrical construction 
 the points where it will strike (i) the horizontal plane passing 
 through the point of projection, (ii) an inclined plane through 
 the same point. 
 
 Compare Art. 90, 
 
350 PEOBLEMS. 
 
 30. Chords are drawn joining any point of a vertical 
 circle with its highest and lowest points; prove that if a 
 heavy particle slide down the latter chord, the parabola, 
 which it will describe after leaving the chord, will be touched 
 by the former chord, — and that the locus of the points of con- 
 tact will be a circle. 
 
 31. If the plane in Art. 89, Dynamics, be a rectangle 
 of given sides, find the velocity with which the particle must 
 be projected from one corner so as to leave the plane hori- 
 zontally at the other corner: and shew that the ratio of the 
 horizontal range after leaving the plane to that described on 
 the plane is the sine of the z of elevation of the plane. 
 
 32. The barrel of a rifle sighted to hit the centre of the 
 bull's-eye which is at the same height as the muzzle and 
 distant a yards from it, would be inclined at an elevation a 
 to the horizon. Prove that if the rifle be wrongly sighted 
 so that the elevation is a + 6, 6 being small compared with a, 
 
 the target will be hit at a height 5— . 6 above the centre 
 
 ° cos a 
 
 of the bull's-eye. 
 
 If the range be 960 yds., the time of flight 2", and the 
 error of elevation 1", the height above the centre of the bull's- 
 eye at which the target will be hit will be nearly ^th of an 
 inch. 
 
 33. A ball of elasticity e is projected obliquely up an 
 inclined plane so that the point of impact at the third time 
 of striking the plane is in the same horizontal line as 
 the point of projection. Prove that the distances from this 
 line of the points of first and second impact are in the 
 ratio 1 : e. 
 
PEOJECTILES. CHAP. IV. 351 
 
 34. If a ball be projected from a point in an inclined 
 plane in a direction such that the range on the plane is the 
 greatest, shew that the direction of motion on striking the 
 plane is perpendicular to the direction of projection. 
 
 35. An imperfectly elastic particle falls down an inclined 
 plane of given length, and at the foot impinges on a hori- 
 zontal plane; shew that the range on this plane will be 
 greatest when the angle of elevation of the inclined plane is 
 = tan~V2. 
 
 36. A body of elasticity e is projected from a point in a 
 
 horizontal plane. If the distance of the point of w* impact be 
 
 equal to four times the sum of the vertical spaces described, 
 
 1 + e . 
 
 5 is the tangent of the angle of projection. 
 
 37. If a be the angle of projection of a projectile, T 
 
 the time which elapses before the body strikes the ground, 
 
 T ... 
 
 Drove that' at the time - — ^-5- the angle which the direction 
 ^ 4 sm a 
 
 of motion makes with the direction of projection is equal to 
 
 2-"- 
 
 38. If three heavy particles be projected simultaneously 
 from the same point in any directions with any velocities, 
 prove that the plane passing through them will always 
 remain parallel to itself. 
 
 39. A perfectly elastic ball is projected from the middle 
 point of one of the sides of an equilateral three-cornered room. 
 It strikes the other two sides and returns to the point of pro- 
 jection. If a be the length of a side of the room and the 
 
352 PKOBLEMS. 
 
 5a 
 velocity of projection be that due to the height — , shew that 
 
 1 3 
 
 the ball must be projected at an angle = - sin"' - . 
 
 40. An elastic ball is let fall from a given height above 
 a smooth inclined plane; shew that the time of making a 
 given number of hops is the same for all inclinations of the 
 plane. 
 
 41. Heavy particles are projected horizontally with dif- 
 ferent velocities from the same point; shew that the extre- 
 mities of the latera recta of the parabolas which they generally 
 describe, lie on a cone, of which the axis is vertical and the 
 vertical angle 2 tan"' 2. 
 
 42. ABC is a right-angled triangle in a vertical plane 
 with its hypotenuse AJB horizontal; a particle projected from 
 A passes through C and falls at B: prove that the tangent 
 of the angle of projection = 2 cosec 2A, and that the latus 
 rectum of the path described is equal to the height of the 
 triangle. 
 
 43. A perfectly elastic particle dropped from a point P 
 impinges upon an inclined plane at Q. If PN be perpen- 
 dicular to the plane, shew that the range = 8 . QN, — and 
 hence find the locus of P in order that the particle may after 
 one reflexion strike a given point in the plane. 
 
 44. A particle A is projected at an angle a to the horizon 
 
 with velocity V, and is met by a second particle B which is 
 
 let fall from the directrix at the instant of projection of A, — 
 
 shew that the distance of the line described by B from the 
 
 vertical line drawn through the point of projection of A is 
 
 V' 
 = -cota. 
 
PROJECJILEIS. CHAP. IV. 353 
 
 45. If rj, j-j, r, be tliree distances of a projectile from 
 the point of projection at which its angular elevations ahove 
 the point of projection are respectively a^, a^, «g — ^shew that 
 
 rj cos" Kj sin (a^ - a^ + r^ eos' a^ sin (a^ - aj 
 
 + r-3 cos" a, sin (a^ — a J = 0. 
 
 46. From several points of a plane superficies inclined 
 to the horizon bodies are projected simultaneously in different 
 directions, in such a manner that the times of flight along tie 
 superficies are the same. Prove that the locus of the bodies 
 at any moment is a plane parallel to the superficies. 
 
 47. Tangents at points P, Q in the parabolic path of a 
 particle acted on by gravity, meet in T. If S be the focus, 
 shew that the velocity due to the height 8T is a mean pro- 
 portional between the velocities at P and Q. 
 
 48. A plane is inclined at an angle of 45° to the horizon, 
 
 and from the foot of it a body is projected upwards along the 
 
 plane, and reaches the top with ith of its original velocity 
 
 iy) ; where will it strike the ground ? 
 
 3 v^ 
 Result. At a distance = ^ — from the point of projection. 
 
 49. A perfectly elastic particle is dropped from a point 
 on the interior surface of a fixed smooth sphere : shew that 
 after its second impact on the sphere it will ascend vertically, 
 and will continually pass and repass along the same vertical 
 and parabolic paths, if the horizontal distance of its first 
 
 vertical path from the centre be \'^Z- V2 a, where a = rad. 
 of sphere. 
 
 50. Two inclined planes of the same altitude h and the 
 same inclination a are placed back to back on a horizontal 
 
 p. M. 23 
 
354 PEOBLEMS. 
 
 plane. A ball is projected from the foot of one plane along 
 its surface and in a direction making an ^ /8 with its hori- 
 zontal edge. After flying over the top of the ridge it falls 
 at the foot of the other plane: shew that the velocity of pro- 
 jection is ^ 
 
 J V^A (8 + cosec" 'a) . cosec ^. 
 
 51. An imperfectly elastic ball is dropped into a hemi- 
 spherical bowl from a height n times the radius of the bowl 
 above the point of impact, so as to strike the bowl at a point 
 30° from its lowest point, and just rebounds over the edge of 
 the bowl : shew that the elasticity of the ball is = V 3 . n~K 
 
 52. An imperfectly elastic particle is projected with a 
 given velocity from a point in a horizontal plane from which 
 it continually rebounds; shew that the sum of the areas of the 
 parabolic segments it will describe will be a maximum when 
 the z of projection is 60°, and that then it is 
 
 _V3 v* 
 
 53. A ball of elasticity e is projected from a point in an 
 inclined plane, and after once impinging upon the inclined 
 plane, rebounds to its point of projection ; prove that, a being 
 the inclination to the horizon of the inclined plane, and /3 that 
 of the direction of projection to the inclined plane, 
 
 cot a . cot /3 = 1 + e. 
 
 54. If a projectile can be shot through three points {a, I), 
 {a, V), (a", 6") in the same vertical plane, prove that 
 
 ab"-a!'h ah' - a'h 
 
 a" (a" — a) a {a! — a) ' 
 the point of projection being the origin and the axis of x 
 horizontal. 
 
PROJECTILES. CHAP. IV. 355 
 
 55. If V, v, v" be the velocities at tliree points P, Q, B 
 of the path of a projectile, where the inclinations to the hori- 
 zon are a, a — /3, a — 2/3, and if t, t' be the times of describing 
 FQ, QR respectively, shew that 
 
 „. ., J ] , 1 2COS/3 
 V t=vt, and - + -^ = ^ — ^ . 
 
 V V V 
 
 56. A bodj is thrown over a triangle, passing from on& 
 extremity of the horizontal base just over the vertex to the 
 other extremity of the base; prove that tan 6 = tan a + tan /3, 
 where is the angle of projection, and a, yS are the angles at 
 the base of the triangle. 
 
 57. From every point in the path of a projectile particles 
 are projected, in the same direction as the projectile at that 
 
 point, and with - th of the velocity, — shew that the locus of 
 
 the foci of the paths described is a parabola. 
 
 58. A number of particles are projected in one vertical 
 plane, from the same point P, so that the foci of their paths 
 shall be in a given straight line not passing through P, and 
 making an angle a. with a horizontal plane. If v -be the 
 velocity, and ^ the angle of projection of any one, shew that 
 v^ cos (a — 2^) is the same for all : and if P/S be perpendicular 
 to the given line, 8 is the focus of the parabola when the 
 
 angle of projection is ^ « 
 
 59. If n equal particles be projected from the same poijit 
 
 with the same velocity v, and in directions making the angles 
 
 a, Sol, 5a, &c. with the horizon, and in the same plane, — provfe 
 
 that their centre of gravity will describe the path of a body 
 
 , . « sin wa 
 projected at an angle m with a velocity - ^ ^-^ ^ • ; 
 
 23—2 
 
356 PROBLEMS, 
 
 60. From a point P on the ground equidistant between 
 two vertical planes A and B, an imperfectly elastic baill is 
 projected with a yelocity = \/{2gh) towards A, and reflected 
 by it to 5 ; find c the altitude of the highest point of B the 
 ball can reach, and shew, 
 
 (I) That if a be the elevation of the direction of pro- 
 jection which enables the ball to attain that altitude, 
 
 J, 
 tan^ a ■■ 
 
 h-c' 
 
 (ii) That if a.', a" be two elevations such that 
 tan a' + tan a" = 2 tan a, 
 two balls projected in those directions towards A will hit the 
 same point of B. 
 
 61. The time of a particle under the action of gravity 
 describing any arc of its parabolic path bounded by a focal 
 chord, is equal to the time of falling from rest vertically 
 through a distance equal to the length of that chord. 
 
 62. An elastic ball is projected in a given manner from 
 a point -4 in a horizontal plane, and at the moment it is 
 moving horizontally it impinges directly upon an equal ball 
 moving in the opposite direction with the same velocity; 
 shew that it will return to A after one rebound if its elas- 
 ticity = ^. 
 
 63. Two elastic balls are projected towards each other in 
 the same vertical plane, v being the velocity and a the angle 
 of projection of each ; shew that after impinging on each other 
 they will return to the points of projection if 
 
 ffa{l + e) =ey'sin2a, 
 
 e being the coefficient of elasticity and 2a the distance be- 
 tween the p(5ints qf projection. 
 
PKOJECTILES. CHAP. IV. 357 
 
 64. Two bodies are projected simultaneously from a point 
 with velocities v, v at elevations a, a ; shew that the time 
 between their passage through the point common to their 
 path is 
 
 _ 2 vv' sin (a ~ a') 
 ff V cos a + v cos a ' 
 
 65. A particle is projected from the vertex of a parabolic 
 tube with velocity due to height h : the axis of the parabola 
 being vertical and vertex downwards ; shew that after quit- 
 ting the tube it will strike the horizontal plane through the 
 vertex in a point whose greatest distance from the vertex is 
 
 where 4a! is the latus rectum. 
 
 Give a geometrical construction for determining the length 
 of the tube for this maximum range. 
 
 Apply the method employed in Art. 92. 
 
 66. A ball whose elasticity is e falls through a vertical 
 height h, and is then reflected by a plane inclined at an 
 angle a to the horizon ; shew that the range on a horizontal 
 plane passing through the point of incidence is 
 
 2h (1 + e) sin 2a (e cos^ a — sin" a). 
 
 Interpret the meaning of this expression when e = 0. 
 
 67. Bodies are projected with the same velocity in dif- 
 ferent directions from the same point A ; the locus of the 
 vertices of the parabolas described is an ellipse whose axis 
 minor is the height due to the velocity of projection, and axis 
 major double the axis minor. 
 
 68. Planes are drawn in every direction from the point 
 A, and bodies are projected from A with given velocity and 
 
358 PROBLEMS. 
 
 at such angles that the ranges on each of these planes shall 
 be the greatest ; shew that the locus of their extremities is 
 •a parabola, which touches the parabolic paths of all the 
 bodies. 
 
 69. A ball projected from a point on an imperfectly 
 elastic horizontal plane strikes a like vertical plane placed at 
 right angles to its direction at the highest point of its tra- 
 jectory. After n rebounds on the horizontal plane it returns 
 to the point of projection, — shew that if e be elasticity 
 
 (l-e)= = 2e''(l-e"). 
 
 70. A plane AB inclined at angle a to the horizon leads 
 up to a horizontal plane BG: a particle is projected from the 
 point A directly up AB, with velocity V, traverses the plane 
 AB, and falls upon the plane BC; — if the times of motion 
 from ^ to -B and from B to Cbe equal, shew that 
 
 . „ _ 27^ sin a (1 + sin' a) 
 
 g (1 + 2 sin' a) 
 
 CURVILINEAR MOTION. CHAPTER V. 
 
 1. If the length of the seconds pendulum be 39'1393 
 inches, find the value of g to three places of decimals. 
 
 2. A clock loses 5" per diem ; how much must its pen- 
 dulum be shortened in order that the error may be corrected, 
 the length of the pendulum being 39 'M inches nearly? 
 
 Result. •0045 inohes nearly. 
 
 3. The force which accelerates a body's motion in a 
 cycloid — whose axis is vertical and vertex downwards — 
 
CURVILINEAR MOTION. CHAP. V. 359 
 
 varies as the arc incepted between the body and the lowest 
 point. 
 
 4. What is the length of a pendulum which vibrates, 
 (i) in J a second, (ii) in J of a second, in the latitude of 
 London ? 
 
 MesvZt. (i) 9-7846 inches, (ii) 2-4462 inches nem-ly. 
 
 5. In a series of experiments made at the Harton coalpit, 
 a pendulum which beat seconds at the surface, gained 2j 
 beats in a day at a depth of 1260 feet : if ff, g be the force 
 of gravity at the surface and at the depth mentioned, shew 
 that 
 
 g 19200 " 
 
 6. How much must a seconds pendulum be shortened in 
 order that it may oscillate seconds on the top of a mountain 
 3000 feet high — assuming the radius of the Earth to be 4000 
 miles, and the force of gravity to vary as (distance)"^ from the 
 centre of the Earth ? 
 
 7. A railway carriage weighing 12 ton's is moving along 
 a circle of radius 720 yards at the rate of 32 miles an hour ; 
 find the horizontal pressure on the rails, or what is commonly 
 called the centrifugal force. 
 
 Result. -39 tons, nearly. 
 
 8. A railway train is going smoothly along a curve of 
 500 yards radius at the rate of 30 miles an hour ; find at what 
 angle a plumb-line hanging in one of the carriages will be 
 inclined to the vertical. 
 
 Result. 2". 14' nearly. 
 
 9. The breadth between the rails in a railway is 4. ft. 
 8J in. Shew that on a curve of 500 yards radius, the outer 
 
360 PROBLEMS. 
 
 rail ought to be raised about 2^ inches for trains travelling 
 30 miles an hour. 
 
 10. A pendulum is found to make 640 vibrations at the 
 equator in the same time as it makes 641 at Greenwich ; if 
 a string hanging vertically can just sustain 80 pounds at 
 Greenwich, how many pounds can the same string sustain 
 at the equator ? 
 
 Result. Atout 80J lbs. 
 
 11. The time of oscillation of a particle in a small arc of 
 a circle is half the time of oscillation in the cycloid which 
 could be generated by the circle. 
 
 12. A seconds pendulum was too long on a given day by 
 a small quantity a, it was then over-corrected so as to be too 
 short by a. during the next day ; shew that the number of 
 
 2 
 
 minutes gained in the two days was 1080 -p^ nearly, if i be 
 the length of the seconds pendulum. 
 
 13. The time of descent to the lowest point in a small 
 circular arc is to the time of descent down its chord = tt : 4. 
 
 14. A perfectly elastic ball is projected obliquely, and on 
 reaching its highest point strikes directly another equal 
 ball hanging by a string from the directrix of its path ; shew 
 that the struck ball will just reach the directrix. 
 
 15. Two particles A, B — of elasticity e — are let fall in 
 opposite directions, at the same instant, from the highest point 
 of a smooth circular tube of very small bore, placed in a 
 vertical position ; find the ratio of their masses in order that 
 
CURVILINEAR MOTION. CHAP. V. 361 
 
 the heavier may remain at rest after impact, and determine 
 the height to which the other will rise. 
 
 Remit. A = (l + 2c) B, and B will' rise to a height =4e'.(iia!me«er', after the 
 unpact. 
 
 16. The attractive force of a mountain horizontally is/, 
 and the force of gravity is g ; shew that the time of vibra- 
 
 tion of a pendulum will be = tt a/ —^ : o being the length 
 
 of the pendulum. 
 
 17. A pendulum which would oscillate seconds at the 
 equator, would if carried to the pole, gain 5' a day ; shew that 
 gravity at the equator : gravity at the pole = 144 : 145. 
 
 18. A railway train is moving smoothly along a curve at 
 the rate of sixty miles an hour, and in one of the carriages 
 a pendulum, which would ordinarily oscillate seconds, is 
 observed to oscillate 121 times in two minutes. Shew that 
 the radius of the curve is very nearly two furlongs. 
 
 Suppose a stone to be dropped from the window of this 
 carriage, find approximately how far from, the rail it will fall. 
 
 19. A particle is suspended by two equal strings from two 
 fixed points in the same horizontal line, the distance between 
 them being equal to the length of either string ; if the particle 
 be raised to one of the fixed points and then dropped, find 
 where it will first come to rest. 
 
 Sesult. When the second string which becomes stretched makes an angle 
 
 e=sin-i — ^ with the horizon. 
 8 
 
 20. A groove is cut alon^ the surface of a right cone of 
 height h, so as always to intersect the generating line at a 
 
362 , PROBLEMS. 
 
 given angle /3 ; shew that the time in which a heavy particle 
 
 will arrive at the base is = k/ \ --^ \ sec a sec /3 : where 
 
 2a is the vertical angle of the cone and \ the vertical dis- 
 tance of the particle from the vertex at the beginning of the 
 motion. 
 
 21. If a heavy particle slide freely from the highest 
 point of a cycloid, of which the axis is vertical and vertex 
 downwards, the angular velocity of the generating circle 
 passing through the point will be constant, — and inversely 
 proportional to the square root of its radius. 
 
 22. A number of cycloids are drawn through a given 
 point A and having their vertices situated on a given curve 
 and their axes vertical. Prove that if the given curve be a 
 cycloid whose vertex is at A and whose axis is vertical, the 
 time of descent from A down all the cycloids to the given 
 curve will be the same : — and that whatever be the form of 
 the given curve the cycloid down which a particle will slide 
 in the greatest or least time will have the tangent at A paral- 
 lel to the tangent drawn to the given curve at the point where 
 the cycloid meets it. 
 
 23. Two unequal weights P, Q are connected by a string 
 of given length (c) which passes through a small ring ; find 
 how many times in a second the lighter one Q must revolve 
 as a conical pendulum, in order that the heavier may be at rest 
 at a given distance a from the ring. 
 
 Result. H- \/ TT, — 7 • times. 
 27r A' Q,{e~a) 
 
 24. Gravity oc ,. r^ ; mass of the Earth = 49 . mass 
 
 of the Moon, and radius of the Earth = 4 . radius of the Moon; 
 
CURVILINEAR MOTION. CHAP. V. 363 
 
 prove that a seconds pendulum carried to the moon would 
 
 7 
 oscillate in - seconds. 
 4 
 
 25. A heavy particle heing projected horizontally from 
 the lowest point of a smooth spherical cavity of radius r, shew 
 that it will never leave the surface of the cavity if the velocity 
 of projection be either < V^ or not < Vs^. 
 
 26. A bead running upon a fine thread, the extremities of 
 which are fixed, describes an ellipse in a plane passing through 
 the extremities, under the action of no external force ; prove 
 that the tension of the thread for any given position of the 
 bead is inversely proportional to the square of the conjugate 
 diameter. 
 
 27. If a particle start from the extremity of the base of 
 a cycloid (as in Art. 102), the velocity at any point will be 
 proportional to the radius of curvature at the point. 
 
 28. Two beads of equal weight are sliding down a per- 
 fectly smooth circular wire in a vertical plane, and are at the 
 same instant at the extremities of a vertical chord subtending 
 a right angle at the centre; find the velocity and direction 
 of motion of their centre of gravity at that instant, each 
 bead having been started from the highest point with an 
 indefinitely small velocity. 
 
 29. A particle is projected from the vertex of a parabolic 
 arc, whose axis is horizontal and plane vertical, up the con- 
 cave side of the arc with a velocity v, and describes an angle 
 20 about the focus before leaving the curve ; shew that 
 
 — =tan'^-h3tan6'. 
 
364 MISCELLANEOUS PROBLEMS 
 
 2c being the length of the latus rectum — and the lengths of 
 the latus rectum of the parabola subsequently described is 
 
 = 2c tan' ^. 
 
 30. A smooth parabola is placed with its axis horizontal 
 and plane vertical, and a particle is projected from the vertex 
 so as to move on the concave side of the curve ; shew that 
 the vertical space described before leaving the curve is two- 
 thirds of the greatest height attained. 
 
 31. A cycloidal arc is placed with its plane vertical, its 
 base horizontal and vertex upwards, and a heavy particle is 
 projected from the cusp up the curve with a velocity due 
 to a height A; shew that the latus rectum of the paTabola 
 
 described after leaving the curve will be — , a being the 
 length of the axis of the cycloid. 
 
 32. A body suspended from a fixed point by a string of 
 length a is projected horizontally from the lowest point with 
 
 velocity = (^3 + 1) A/ y 5 shew that it will pass through the 
 
 point of suspension, and that its direction of motion at that 
 point will make an ^ cos"' J with the horizon. 
 
 MISCELLANEOUS PROBLEMS IN DYNAMICS. 
 
 1. li R, R' be the ranges of the two projectiles, which 
 being thrown from the same place, attain the same vertical 
 height, and pass through a common point, — ^then will 
 
 n 
 
IN DYNAMICS. 365 
 
 where S is the greatest height attained, and h, k are co-ordi- 
 nates of the point common to the two paths. 
 
 2. From a number of points, bodies subject to gravity 
 are projected, all directed towards one point with velocities 
 proportional to the distances of the points of projection from 
 that point. All hit another point. Shew that the points of 
 projection lie in a conic section. 
 
 • 3. Two bevilled wheels roll together ; having given the 
 angular velocity a> of the first wheel and the inclination (a) of 
 the axes of the cones, find their vertical angles that the second 
 wheel may revolve with a given angular velocity w'. 
 
 Jtesult. If 28, 20 be vertioal angles of the first and second wheels, -we 
 must have 
 
 + <p=a., and usin 9=<i)'sin 0. 
 
 4. The highest point of the wheel of a carriage, rolling 
 on a horizontal road, moves twice as fast as each of two 
 points in the rim, whose distance from the ground is half the 
 radius of the wheel. 
 
 5. A ball projected with a velocity v would penetrate 
 into a block of wood m feet ; what velocity would it lose in 
 passing through a board n feet thick, the resistance being 
 uniform ? 
 
 Beiidt. «'fl-\/-^j- 
 
 6. A ball is thrown vertically down on a horizontal 
 pavement, and just rebounds to its original height. Shew 
 that the velocity of projection is to that due to the original 
 height above the pavement as tan {cos~'-e) : 1— e being the 
 elasticity at impact. 
 
366 MISCELLANEOUS PKOBLEMS 
 
 7. A particle is projected up a rough inclined plane; 
 shew that if t^ = time of ascending, t^ = time of descending, we 
 shall hare 
 
 \tj sin (a + 0) ' 
 if the coefficient of friction = tan <^, 
 
 8. Two balls are moving in the same straight line, one 
 of them only being acted on by a force ; if the force be con- 
 stant and tend towards the other ball, shew that the times 
 which elapse between consecutive impacts decrease in geome-- 
 trical progression, 
 
 9. A point moves in such a manner that the sum of the 
 squares of its distances from any number of given points in 
 the same plane with it is constant. Prove that if perpendicu- 
 lars from the points be at any time let fall on its direction of 
 motion, the point itself will be the centre of gravity of the 
 feet of these perpendiculars. 
 
 10. The curve y'-y .f{x)+a?=Q is such that the 
 times down the chords from the origin to any two points in 
 it vertically below each other are the same; the axis of x 
 being horizontal and that of y A'ertical. 
 
 11. Shew that the time of quickest descent from any 
 point of an ellipse to the horizontal axis major down the 
 
 normal is = W — , I being the latus rectum, e the eccen- 
 tricity. 
 
 12. Shew that the circumferences of two circles contain 
 all points from which the time of quickest descent to a given 
 vertical circle is the same. 
 
IN DYNAMICS. 367 
 
 13. A ball whose elasticity is ^ projected from the floor 
 of a room 12 feet high, strikes the ceiling and floor and just 
 rises to the ceiling again, — find the velocity of projection. 
 
 Sesult. J-61i.g, 
 
 14. A perfectly elastic ball is thrown into a smooth cylin- 
 drical well from a point in' the circumference of the circular 
 mouth. Shew that if the ball be reflected any number of 
 times from the surface of the cylinder, the intervals between 
 the reflexions will be equal. 
 
 In the last question, if the ball be projected horizontally, 
 
 rrr 
 
 making an angle — with the tangent at the point of projection, 
 
 it will reach the surface of the water at the instant of the 
 n* reflexion, if the space due to the velocity of projection be 
 
 _ (radius)" 
 
 depth ' V n 
 ******* 
 
 15. From a point T two tangents are drawn to touch a 
 circle in the points P, Q : given that the velocity acquired 
 by a body sliding down the chord PQ is equal to 1-m"' of 
 the velocity down the vertical diameter of the circle, prove 
 that the locus of T is the curve whose equation is 
 
 the centre of the circle being the pole, and (a) the radius, 
 
 16. One end of a string is attached to an angular point 
 of a fixed regular polygon of n sides, its length being equal to 
 the perimeter c ; a particle, attached to the other end of the 
 string which is stretched in direction of a side, is projected in 
 
368 MISCELLANEOUS PEOBLEMS 
 
 the plane of the polygon perpendiculaarly to the string with a 
 given velocity V. Determine after what time the string will 
 coincide with the perimeter of the polygon (the action of gra- 
 vity being neglected). 
 
 Deduce the time when, the perimeter remaining the same, 
 the number of the sides is infinitely increased. 
 
 Beault. . ^=r . 
 
 n V 
 
 17. Explain the object and advantages of rifling the 
 barrel of a gun. 
 
 18. Find the amount of worh done in drawing up a 
 Venetian blind. How must the same problem be solved for 
 a curtain? 
 
 19. A ship is sailing with a uniform velocity in a southerly 
 direction, and is fired upon at the instant it is due east of a 
 battery; given the velocity of a cannon-ball, determine at 
 what elevation and towards what point of the compass it 
 must be fired that it may strike the ship. 
 
 20. Two perfectly elastic balls are dropped from two 
 points not in the same vertical line, and strike against a per- 
 fectly elastic horizontal plane ; shew that their centi'e of gravity 
 will never reascend to its original height, unless the initial 
 heights of the balls be in the ratio of two square numbers. 
 
 21. A smooth tube of uniform bore and radius a, is bent 
 into the form of a circular arc — (= 27r — 2a) — greater than a 
 semicircle, and placed in a vertical plane with its open ends 
 upwards, and in the same horizontal line. Find the velocity 
 u with which a ball that fits the tube must be projected along. 
 
IN DYNAMICS. 369 
 
 the interior from the lowest point, in order that it may pass 
 out at one end and re-enter at the other. 
 
 Reavlt. «'' = jra (2 + 2 cos a + aec o) . 
 
 22. A body P lying on a table is connected with another 
 Q by a string passing over a puUy directly over P; if Q fall 
 through a given height before the string becomes tight, deter- 
 mine the impulsive tension of the string when that takes 
 place, and the change of velocity of Q. 
 
 Compare Art. 75, 76. 
 
 23. Two particles start simultaneously from the same 
 point and move along two straight lines, the one with uni- 
 form velocity, the other from rest with uniform acceleration. 
 Prove that the line joining the particles at any time is always 
 a tangent to a fixed parabola. 
 
 24. If G be the centre of curvature corresponding to 
 any point P of the path of a projectile — prove that the ver- 
 tical velocity of G will be proportional to the time elapsed 
 since P was at the highest point of its path. 
 
 25. Several bodies are projected from the same point ^ 
 in different directions with the same velocity ; shew that the 
 locus of them all at any time is a sphere, and find the radius 
 of the sphere and the position of its centre at any time, 
 
 26. A given weight descending vertically draws another 
 up a smooth inclined plane by a string passing over the vertex 
 of the plane. Find the path of their centre of gravity when 
 the bodies move from rest. 
 
 Result. A straight line. 
 
 P. M. 24 
 
370 MISCELLANEOUS PE0BLEM8 
 
 27. Tangents are drawn to a vertical circle, — find the 
 locus of points in them from which particles would descend 
 in straight lines to the centre in the shortest time. 
 
 28. From what height must a perfectly elastic ball 
 be let fall into a fixed hemispherical bowl, in order that it 
 may rebound horizontally at the first impact and strike the 
 lowest point of the bowl at the second ? 
 
 29. From a given height a perfectly elastic particle is let 
 fall on a perfectly hard inclined plane, so as to strike it at 
 a given fixed point : prove that whatever be the inclination 
 of the plane to the horizon, the vertex of the parabola which 
 the particle describes after impact will lie in a certain 
 ellipse. 
 
 30. A perfectly elastic ball is projected from the foot of 
 one of the walls of a room, against the opposite wall, in a 
 vertical plane perpendicular to both the walls ; shew that if 
 it be required to hit the ceiling after the rebound, the ball 
 must strike the wall at a point at least f ths of the height of 
 the room from the floor. 
 
 31. A rigid wire without appreciable mass is formed 
 into an arc of an equiangular spiral, and carries a small heavy 
 particle fixed in its pole. If the convexity of the wire be 
 placed in contact with a perfectly rough horizontal plane, 
 prove that the point of contact with the plane will move with 
 uniform acceleration — and find this acceleration. 
 
 32. AA, BB are the axes of an ellipse. A smootli 
 tube is bent into the shape of the portion AB'A'B, the ends 
 A, B being open, and the tube is held with B' on a .given 
 horizontal plane. A particle is dropped from a certain height 
 
IN DYNAMICS. 371 
 
 into the tube at A so that after emerging at B it again enters 
 at A. The tube is then held with A' on the horizontal plane 
 and the particle is dropped from the same point so as to fall 
 into the tube at £, and it is found that after emerging at A, 
 it again enters at B. Prove that the eccentricity of the ellipse 
 
 ■ V45(3-V5) 
 IS . 
 
 33. If two parabolas be placed with their axes vertical, 
 vertices downwards and foci coincident, prove that there are 
 three chords down which the time of descent of a particle 
 under the action of gravity from one curve to the other is 
 a minimum ; — and that one of these is the principal diameter 
 and the other two make an angle of 60° with it on either side. 
 ******* 
 
 34 A ball thrown from any point in one of the walls of 
 a rectangular room after striking the three others returns to 
 the point of projection before it falls to the ground. Shew 
 that the space due to the velocity of projection is greater than 
 the diagonal of the floor. 
 
 35. There are three equal and perfectly elastic balls 
 A, B, C. A is let fall from a given point, and at a moment 
 when it reaches a given horizontal plane, B is let fall from the 
 same point, and at the moment when A in returning meets B, 
 G is let fall. Shew that B will meet for the second time 
 where it first met A. 
 
 36. Two planes having a common altitude h are inclined 
 
 at angles a and ^ - a to the horizon; two equal, indefinitely 
 
 small and perfectly elastic balls are projected along them with 
 equal velocities Ffrom their feet, and so that they may im- 
 
372 MISCELLANEOUS PROBLEMS 
 
 pinge at tlie top ; shew that if the ball which ascends along 
 the former plane falls at its foot after impact, then 
 
 -V =(1 + cot a) + (1 + cot a)-\ 
 
 37. There are generally two directions in which a pro- 
 jectile may he projected with given velocity from a point A, 
 so as to pass through another point B ; and one of these di- 
 rections is inclined to the vertical at the same angle that the 
 other is inclined to the line AB. Hence shew that the locus 
 of points, for which a given sight must he used in firing with 
 a given charge of powder, is the surface generated by the 
 revolution, about the vertical, of the path of the bullet obtained 
 by aiming at the zenith with the given sight, and the given 
 charge of powder. 
 
 See Solutions of Senate-House Problems for 1854, Walton and Mackenzie, 
 p. 33. 
 
 38. A perfectly elastic particle projected against one side 
 of a plane polygon is reflected at the other sides in succes- 
 sion, the polygon being such that the angle of incidence 
 on each side is the same; find the impulse on the particle 
 at each impact, and deduce the expression for the normal 
 
 pressure f - 1 on a particle moving freely on a curve under the 
 
 actioii of no other impressed force. 
 
 39. A body falls from rest under the action of an 
 accelerating force which remains constant during certain 
 successive equal intervals of time, but is changed at the 
 expiration of each such interval so that the space described 
 
 in the n*'' interval is always times the space described 
 
IN DYNAMICS. 373 
 
 in tlie first of them. If the velocity acquired at the end of 
 the first interval be v, shew that after a long lapse of time 
 the velocity approaches a uniform velocity 2v. 
 
 40. Three smooth equal perfectly elastic billiard balls 
 A, B, C are placed with their centres in the angular points 
 of an equilateral triangle ; shew that it will be impossible, 
 with another equal ball, to cannon off A on to B, — A itself 
 striking C, — unless the diameter of each ball be equal to half 
 a side of the triangle. 
 
 41. A particle of given elasticity e is projected down a 
 smooth vertical cylinder of indefinite length, but terminated 
 by a horizontal plane at its lower end ; the particle initially 
 remaining in contact with the cylinder. If it be projected at 
 a height h from the bottom with velocity F at an ^ a with 
 the vertical, then after the time 
 
 ^2gh+ Pcos'g 1 + e 
 ~9 •!-«' 
 
 it will be moving uniformly with velocity Fsin a. 
 
 42. Two perfectly elastic balls A and B impinge upon 
 each other. First A impinges upon B at rest and goes ofi" in 
 a direction making an ^ ^ with the line joining their centres : 
 then B impinges upon A at rest and at the same angle of 
 incidence, and goes off at an ^ 6'. Prove that 6 + & = 180°. 
 
 Prove also that if the balls be imperfectly elastic, and the 
 angles of incidence in the two cases be a and a', then 
 
 cot 6 cot ff , 
 
 h ■ ; = 1 - e. 
 
 cot a cot a 
 
 43. Two equal balls, one perfectly elastic, the other 
 inelastic, are dismissed by the same horizontal blow from the 
 
374 MISCELLANEOUS PROBLEMS IN DYNAMICS. 
 
 top of a flight of uniform steps, so that each falls just on the 
 margin of the first step : shew that the number of steps 
 cleared by the elastic ball in its successive flights is the 
 series of successive odd numbers, — and that the two balls 
 reach the bottom of the steps simultaneously. 
 
 44. From a point in the lower one of two parallel 
 horizontal planes a ball of elasticity e is projected at an 
 angle a, — is reflected by the upper plane, and again reflected 
 by the lower one; the distance between the planes being 
 
 f-jth that due to the velocity V of projection. liv be the 
 
 velocity of the ball in rebounding for the «i"' time from the 
 lower plane. 
 
 v'^V'(> 
 
 cos''a + e'"sin''a + - 
 
 n 1 
 
 THE END. 
 
CAMBRIDGE : 
 PEINTED AT THE UNIVKBSIIT PRESS. 
 
By the same Author. 
 
 A TREATISE ON OPTICS. 
 
 304 pp. (1859). Crown 8vo. los. 6d. 
 
 The present work may he regarded as a new edition of the Treatise on 
 Optics, by the Rev. W. N. Griffin, which being some time ago out of print, 
 was very kindly and liberally placed at the disposal of the author. The author 
 has freely used the liberty accorded to him, and has rearranged the matter 
 with considerable alterations and additions — especially in those parts which 
 required more copious explanation and illustration to render the work suitable 
 for the present course of reading in the University. A collection of Examples 
 and Problems has been appended, which are sufficiently numerous and varied 
 in character to affiird an useful exercise for the student : for the greater part of 
 them recourse has been had to the Examination Papers set in the University 
 and the several Colleges during the last twenty years. 
 
 Subjoined to the copious Table of Contents the author has ventured to indi- 
 cate an elementary course of reading not unsuitable for the requirements of the 
 First Three Days in the Cambridge Senate-House Examinations.